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+{"text":"\\section{#1}\n}\n\\def\\hskip 1pt\\vrule  width1pt\\hskip 1pt{\\hskip 1pt\\vrule  width1pt\\hskip 1pt}\n\\def{\\mathsf b}{{\\mathsf b}}\n\\def\\mathsf P{\\mathsf P}\n\\def{\\mathbb H}{{\\mathbb H}}\n\\def{\\mathbb N}{{\\mathbb N}}\n\\def{\\mathbb R}{{\\mathbb R}}\n\\def{\\mathbb C}{{\\mathbb C}}\n\\def{\\mathbb Z}{{\\mathbb Z}}\n\\def{\\mathbb P}{{\\mathbb P}}\n\\def{{\\rm I}\\kern-1pt\\Pi}{{{\\rm I}\\kern-1pt\\Pi}}\n\\def{\\mathbb S}{{\\mathbb S}}\n\\def\\mathfrak{S}{\\mathfrak{S}}\n\\def\\mathbb T{\\mathbb T}\n\\def\\mathcal{T}{\\mathcal{T}}\n\\def\\alpha{\\alpha}\n\\def\\b #1;{{\\bf #1}}\n\\def{\\bf x}{{\\bf x}}\n\\def{\\bf k}{{\\bf k}}\n\\def{\\bf y}{{\\bf y}}\n\\def\\mathbf{u}{\\mathbf{u}}\n\\def{\\bf w}{{\\bf w}}\n\\def{\\bf z}{{\\bf z}}\n\\def{\\bf h}{{\\bf h}}\n\\def\\mathfrak{m}{\\mathfrak{m}}\n\\def\\mathbf{j}{\\mathbf{j}}\n\\def\\mathbf{r}^*{\\mathbf{r}^*}\n\\def{\\mathbf 0}{{\\mathbf 0}}\n\\def\\epsilon{\\epsilon}\n\\def\\lambda{\\lambda}\n\\def\\phi{\\phi}\n\\def\\varphi{\\varphi}\n\\def\\sigma{\\sigma}\n\\def\\mathbf{t}{\\mathbf{t}}\n\\def\\Delta{\\Delta}\n\\def\\delta{\\delta}\n\\def{\\cal F}{{\\cal F}}\n\\def{\\cal O}{{\\cal O}}\n\\def{\\cal P}{{\\cal P}}\n\\def{\\mathcal M}{{\\mathcal M}}\n\\def{\\mathcal C}{{\\mathcal C}}\n\\def\\CR{{\\mathcal C}}\n\\def{\\mathcal N}{{\\mathcal N}}\n\\def{\\cal Z}{{\\cal Z}}\n\\def{\\cal A}{{\\cal A}}\n\\def{\\cal V}{{\\cal V}}\n\\def{\\cal E}{{\\cal E}}\n\\def{\\mathcal H}{{\\mathcal H}}\n\\def{\\bf W}{{\\bf W}}\n\\def{\\mathbb W}{{\\mathbb W}}\n\\def\\mathbb{Y}{\\mathbb{Y}}\n\\def{\\bf Y}{{\\bf Y}}\n\\def{\\widetilde\\epsilon}{{\\widetilde\\epsilon}}\n\\def{\\widetilde E}{{\\widetilde E}}\n\\def{\\overline{B}}{{\\overline{B}}}\n\\def\\ip#1#2{{\\langle {#1}, {#2}\\rangle}}\n\\def\\derf#1#2{{#1}^{(#2)}}\n\\def\\node#1#2{x_{{#1},{#2}}}\n\\def\\mathop{\\hbox{{\\rm ess sup}}}{\\mathop{\\hbox{{\\rm ess sup}}}}\n\\def{{\\bf e}_{q+1}}{{{\\bf e}_{q+1}}}\n\\def\\begin{equation}{\\begin{equation}}\n\\def\\end{equation}{\\end{equation}}\n\\def\\begin{eqnarray}{\\begin{eqnarray}}\n\\def\\end{eqnarray}{\\end{eqnarray}}\n\\def\\eref#1{(\\ref{#1})}\n\\def\\displaystyle{\\displaystyle}\n\\def\\cfn#1{\\chi_{{}_{ #1}}}\n\\def\\varrho{\\varrho}\n\\def\\intpart#1{{\\lfloor{#1}\\rfloor}}\n\\def\\largeint#1{{\\lceil{#1}\\rceil}}\n\\def\\mathbf{ \\hat I}{\\mathbf{ \\hat I}}\n\\def\\mbox{{\\rm dist }}{\\mbox{{\\rm dist }}}\n\\def\\mbox{{\\rm Prob }}{\\mbox{{\\rm Prob }}}\n\\def\\donchitre#1#2{\\vskip 6.5cm\\noindent\n\\parbox[t]{1in}{\\special{eps:#1.eps x=6.5cm y=5.5cm}}\n\\hbox to 7cm{}\\parbox[t]{0.0cm}{\\special{eps:#2.eps x=6.5cm y=5.5cm}}}\n\\def\\chitra#1{\\vskip 9.5cm\\noindent\n\\hskip2in{\\special{eps:#1.eps x=8.5cm y=8.5cm}}\n}\n\\defFrom\\ {From\\ }\n\\def|\\!|\\!|{|\\!|\\!|}\n\\def{\\mathbb X}{{\\mathbb X}}\n\\def{\\mathbb B}{{\\mathbb B}}\n\\def\\mbox{\\mathsf{span }}{\\mbox{\\mathsf{span }}}\n\\def\\bs#1{{\\boldsymbol{#1}}}\n\\def\\bs{\\omega}{\\bs{\\omega}}\n\\def\\bs{\\theta}{\\bs{\\theta}}\n\\def\\mathbf{t}^*{\\mathbf{t}^*}\n\\def\\mathbf{x}^*{\\mathbf{x}^*}\n\\def\\mathsf{dist }{\\mathsf{dist }}\n\\def\\mathsf{supp\\ }{\\mathsf{supp\\ }}\n\\def\\corr#1{{\\color{red} {#1}}}\n\\title{Deep nets for local manifold learning}\n\\author{Charles K. Chui\\thanks{Department of Statistics, Stanford  University, Stanford, CA 94305. The research of this author is supported by ARO Grant W911NF-15-1-0385.\n\\textsf{email:} ckchui@stanford.edu. }\\ \n\\  and\nH.~N.~Mhaskar\\thanks{Department of Mathematics, California Institute of Technology, Pasadena, CA 91125;\nInstitute of Mathematical Sciences, Claremont Graduate University, Claremont, CA 91711. The research of this author is supported in part by ARO Grant W911NF-15-1-0385.\n\\textsf{email:} hrushikesh.mhaskar@cgu.edu. }       }\n\\date{}\n\\begin{document}\n\\maketitle\n\\begin{abstract}\nThe problem of extending a function $f$ defined on a training data ${\\mathcal C}$ on an\nunknown manifold ${\\mathbb X}$ to the entire manifold and a tubular neighborhood of this manifold is considered in this paper.  For ${\\mathbb X}$ embedded in a high dimensional ambient Euclidean space ${\\mathbb R}^D$,  a deep learning algorithm is developed for finding a local coordinate system for the manifold \\textbf{without eigen--decomposition}, which reduces the problem to the classical problem of function approximation on a low dimensional cube. Deep nets (or multilayered neural networks) are proposed to accomplish this approximation scheme by using the training data. Our methods do not involve such optimization techniques as back--propagation, while assuring optimal (a priori) error bounds on the output in terms of the number of derivatives of the target function. In addition, these methods are universal, in that they do not require a prior knowledge of the smoothness of the target function, but adjust the accuracy of approximation locally and automatically, depending only upon the local smoothness of the target function. Our ideas are easily extended to solve both the pre--image problem and the out--of--sample extension problem, with a priori bounds on the growth of the function thus extended. \n\\end{abstract}\n\n\\bhag{Introduction}\nMachine learning is an active sub--field of Computer Science on algorithmic development for learning and making predictions based on some given data, with a long list of applications that range from computational finance and advertisement, to information retrieval, to computer vision, to speech and handwriting recognition, and to structural healthcare and medical diagnosis. In terms of function approximation, the data for learning and prediction can be formulated as $\\{({\\bf x}, f_{\\bf x})\\}$, obtained with an unknown probability distribution. Examples include: the Boston housing problem (of predicting the median price $f_{\\bf x}$ of a home based on some vector ${\\bf x}$ of 13 other attributes \\cite{vapnik2013nature}) and the floor market problem \\cite{tiao1989model, chakraborty1992forecasting} (that deals with the indices of the wheat floor pricing in three major markets in the United States). For such problems, the objective is to predict the index $f_{\\bf x}$ in the next month, say, based on a vector ${\\bf x}$ of their values over the past few months. Other similar problems include the prediction of blood glucose level $f_{\\bf x}$ of a patient based on a vector ${\\bf x}$ of the previous few observed levels \\cite{sergei, mnpdiabetes}, and the prediction of box office receipts ($f_{\\bf x}$) on the date of release  of a movie in preparation, based on a vector ${\\bf x}$ of the survey results about the movie \\cite{sharda2002forecasting}.  It is pointed out in \\cite{multilayer, mauropap, compbio} that all the pattern classification problems can also be viewed fruitfully as problems of function approximation. While it is an ongoing research to allow non--numeric input ${\\bf x}$ (e.g., \\cite{treepap}), we restrict our attention in this paper to the consideration of ${\\bf x}\\in{\\mathbb R}^D$, for some integer $D\\ge 1$.\n\n\nIn the following discussion, the first component ${\\bf x}$ is considered as input, while the second component $f_{\\bf x}$ is considered the output of the underlying process. The central problem is to estimate the conditional expectation of $f_{\\bf x}$ given ${\\bf x}$. Various statistical techniques and theoretical advances in this direction are well--known (see, for example \\cite{vapnik1998statistical}). In the context of neural and radial--basis--function networks, an explicit formulation of the input\/output machines was pointed out in \\cite{girosi1990networks,girosi1995regularization}. More recently, the nature of deep learning as an input\/output process is formulated in the same way, as explained in \\cite{lecun2015deep, poggio_deep_net_2015}. To complement the statistical perspective and understand  the theoretical capabilities of these processes, it is customary to think of the expected value of $f_{\\bf x}$,  given ${\\bf x}$ , as a function $f$ of ${\\bf x}$. The question of empirical estimation in this context is to carry out the approximation of $f$ given samples $\\{({\\bf x}, f({\\bf x}))\\}_{{\\bf x}\\in{\\mathcal C}}$, where ${\\mathcal C}$ is a finite \\emph{training data} set. In practice, because of the random nature of the data, it may be possible that there are several pairs of the form $({\\bf x}, f_{\\bf x})$ in the data for the same values of ${\\bf x}$. In this case, a statistical scheme, such as some kind of averaging of $f_{\\bf x}$ being the simplest one, can be used to obtain a desired value $f({\\bf x})$ for the sample of $f$ at ${\\bf x}$, ${\\bf x}\\in{\\mathcal C}$. From this perspective, the problem of extending $f$ from the traning data set ${\\mathcal C}$ to ${\\bf x}$ not in ${\\mathcal C}$ in machine  learning is called the \\emph{generalization problem}.\n\nWe will  illustrate this general line of ideas by using neural networks as an example. To motivate this idea, let us first recall a theorem originating with Kolmogorov and Arnold \\cite[Chapter~17, Theorem~1.1]{lorentz_advanced}. According to this theorem, there exist \\emph{universal} Lipschitz continuous functions $\\phi_1,\\cdots,\\phi_{2D+1}$ and \\emph{universal} scalars $\\lambda_1,\\cdots,\\lambda_D\\in (0,1)$, for which every continuous function $f : [0,1]^D\\to {\\mathbb R}$ can be written as \n\\begin{equation}\\label{kolmtheo}\nf({\\bf x})=\\sum_{j=1}^{2D+1}g\\left(\\sum_{k=1}^D \\lambda_k\\phi_j(x_k)\\right), \\qquad {\\bf x}=(x_1,\\cdots,x_D)\\in [0,1]^D,\n\\end{equation}\nwhere $g$ is a continuous function that depends on $f$. In other words, for a given $f$, only one function $g$ has to be determind to give the representation formula \\eref{kolmtheo} of $f$. \n\nA neural network, used as an input\/output machine, consists of an input layer, one or more hidden layers, and an output layer. Each hidden layer consists of a number of neurons arranged according to the network architecture. Each of these neurons has a local memory and performs a simple non--linear computation upon its input. The input layer fans out the input ${\\bf x}\\in{\\mathbb R}^D$ to the neurons at the first hidden layer. The output layer typically takes a linear combination of the outputs of the neurons at the last hidden layer. \nThe right hand side of \\eref{kolmtheo} is a neural network with two hidden layers. The first contains $D$ neurons, where the $j$--th neuron computes the sum $\\sum_{k=1}^D \\lambda_k\\phi_j(x_k)$. The next hidden layer contains $2D+1$ neurons each evaluating the function $g$ on the \noutput of the $j$--th neuron in the first hidden layer. The output layers takes the sum of the results as indicated in \\eref{kolmtheo}.\n\n\nFrom a practical point of view, such a network is clearly hard to construct, since only the existence of the functions $\\phi_j$ and $g$ is known, without a numerical procedure for computing these. In the early mathematical development of neural networks during the late 1980s and early 1990s, instead of finding these functions for the representation of a given continuous function $f$ in \\eref{kolmtheo}, the interest was to study the existence and characterization of  \\emph{universal} functions $\\sigma :{\\mathbb R}\\to{\\mathbb R}$, called \\emph{activation functions} of the neural networks, such that each neuron evaluates the activation function upon an affine transform of its input, and the network is capable of approximating any desired real-valued continuous target function $f: K\\to{\\mathbb R}$ arbitrarily closely on $K$, where $K\\subset {\\mathbb R}^D$ is any compact set.\n\nFor example, a neural network with one hidden layer can be expressed as a function\n\\begin{equation}\\label{onehiddenlayer}\n\\mathcal{N}({\\bf x})=\\mathcal{N}_n(\\{{\\bf w}_k\\}, \\{a_k\\}, \\{b_k\\};{\\bf x})= \\sum_{k=1}^n a_k\\sigma({\\bf w}_k\\cdot {\\bf x} +b_k), \\qquad {\\bf x}\\in{\\mathbb R}^D.\n\\end{equation}\nHere, the hidden layer consists of $n$ neurons, each of which has a local memory. The local memory of the $k$--th neuron contains the \\emph{weights} ${\\bf w}_k\\in {\\mathbb R}^D$, and the \\emph{threshold} $b_k\\in{\\mathbb R}$. Upon receiving the input ${\\bf x}\\in{\\mathbb R}^D$ from the input later, the $k$--th neuron evaluates $\\sigma({\\bf w}_k\\cdot{\\bf x}+b_k)$ as its output, where $\\sigma$ is a non--linear activation function. The output layer is just one circuit where the coefficients  $\\{a_k\\}$ are stored in a local memory, and the evaluates the linear combination as indicated in \\eref{onehiddenlayer}.  Training of this network in order to learn a function $f$ on a compact subset $K\\subset{\\mathbb R}^D$ to an accuracy of $\\epsilon>0$ involves finding the parameters $\\{a_k\\}$, $\\{{\\bf w}_k\\}$, $\\{b_k\\}$ so that \n\\begin{equation}\\label{universal appr} \n\\max_{{\\bf x}\\in K}|f({\\bf x})-\\mathcal{N}({\\bf x})|<\\epsilon.\n\\end{equation}\nThe most popular technique for doing this is the so called back--propagation, which seeks to find these quantities by minimizing an error functional usually with some regularization parameters. \nWe remark that  the number $n$ of \\emph{neurons} in the approximant \\eref{onehiddenlayer} must increase, if the tolerance $\\epsilon >0$ in the approximation\nof the target function $f$ is required to be smaller.    \n\n\nFrom a theoretical perspective, the main attraction of neural networks with one hidden layer is their \\emph{universal approximation property} as formulated in \\eref{universal appr}, which overshadows the properties of their predecessors, namely: the perceptrons \\cite{minsky1988perceptrons}. In particular, the question of finding sufficient conditions on the actvation function $\\sigma$ that ensure the universal approximation property was investigated in great detail by many authors, with emphasis on the most popular \\emph{sigmoidal function}, defined by the property $\\sigma(t)\\to 1$ for $t\\to\\infty$ and $\\sigma(t)\\to 0$ for $t\\to -\\infty$. For example,  Funahashi \\cite{funahashi1989} applied some discretization of an integral formula from \\cite{irie1988} to prove the universal approximation property for some sigmoidal function $\\sigma $.  A similar theorem was proved by Hornik, Stinchcombe, White \\cite{hornik1989} by using the Stone--Weierstrass theorem, and another by Cybenko \\cite{cybenko1989} by applying the Hahn--Banach and Riesz Representation theorems.  A constructive proof via approximation by ridge functions  was given in  our paper \\cite{chuili1992}, with algorithm for implementation presented in our follow-up work \\cite{chui1993realization}. A complete characterization of which activation functions are allowed to achieve the universal approximation property was given later in \\cite{mhasmich, leshnolinpinkus}. \n\nHowever, for neural networks with one hidden layer, one of the severe limitations to applying training algorithms based on optimization, such as back--propagation or those proposed in the book \\cite{vapnik1998statistical} of Vapnik, is that it is neccessary to know the number of neurons in $\\mathcal{N}$ in advance. \nTherefore, one major problem in the 1990s, known as the \\emph{complexity problem}, was to estimate the number of neurons required to approximate a function to a desired accuracy. In practice, this gives rise to a trade-off: to achieve a good approximation, one needs to have a large number of neurons, which makes the implementation of the training algorithm harder.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nIn this regard, nearly a century of research in approximation theory suggests that the higher the order of smoothness of the target function, the smaller the number of neurons should be, needed to achieve the desired accuracy. There are many different definitions of smoothness that give rise to different estimates. For example, under the condition that the Fourier transform of the target function $f$ satisfies $\\displaystyle\\int_{{\\mathbb R}^D} |\\bs{\\omega}\\hat{f}(\\bs{\\omega})|d\\bs{\\omega}<\\infty$, Barron  \\cite{barron1993} proved the existence of a neural network with ${\\cal O}(\\epsilon^{-2})$ neurons that gives  an $L^2([0,1]^D)$ error of ${\\cal O}(\\epsilon)$. While it is interesting to note that this number of neurons is essentially independent of the dimension $D$, the constants involved in the ${\\cal O}$ term as well as the number of derivatives needed to ensure the condition on the target function may increase with $D$. Several authors have subsequently improved upon such results under various conditions on the activation function as well as the target function so as to ensure that the constants depend polynomially on $D$ (e.g., \\cite{kurkova1, kurkova2, tractable} and references therein).\n\nThe most commonly understood definition of smoothness is just the number of derivatives of the target function. It is well--known from the theory of $n$-widths that if $r\\ge 1$ is an integer, and the only a priori information assumed on the unknown target function  is that it is $r$--times continuously differentiable function, then a stable and uniform approximation to within $\\epsilon$ by neural networks must have at least a constant multiple of $\\epsilon^{-D\/r}$ neurons.  \nIn \\cite{optneur}, we gave an explicit construction for a neural network  that achieves the accuracy of $\\epsilon$ using ${\\cal O}(\\epsilon^{-D\/r})$ neurons arranged in a single hidden layer. It follows that this suffers from a curse of dimensionality, in that the number of neurons increases exponentially with the input dimension $D$. Clearly, if the smoothness $r$ of the function increases linearly with $D$, as it has to in order to satisfy the condition in \\cite{barron1993}, then this bound is also ``dimension independent''.\n\nWhile this is definitely unavoidable for neural networks with one hidden layer, the most natural way out is to achieve local approximation; i.e., given an input ${\\bf x}$, construct a network with a uniformly bounded number of neurons that approximates the target function with the optimal rate of approximation near the point ${\\bf x}$, preferably using the values of the function also in a neighborhood of ${\\bf x}$. Unfortunately, this can never be achieved as we proved in \n\\cite{chui1994neural}. Furthermore,  we have proved in \\cite{ chui1996limitations} that even if we allow each neuron to evaluate its own activation function, this local approximation fails. Therefore the only way out is to use a neural network with more than one hidden layer, called \\emph {deep net} (for deep neural network). Indeed, local approximation can be achieved by a deep net as proved in our papers \\cite{multilayer, mhaskar1993neural}. In this regard, it is of interest to point out that  an adaptive version of \\cite{multilayer, mhaskar1993neural} was derived in \\cite{lermont} for prediction of time series, yielding as much as 150\\% improvement upon the state--of--the--art at that time, in the study of the floor market problem. \n\nOf course, the curse of dimensionality is inherent to the problem itself, whether with one or more hidden layers. Thus, while it is possible to construct a deep net to approximate a function at each point arbitrarily closely by using a uniformly bounded number of  neurons, the uniform approximation on an entire compact set, such as a cube, would still require an approximation at a number of points in the cube, and this number increasing exponentially with the input dimension. Equivalently, the effective number of neurons for approximation on the entire cube is still exponentially increasing with the input dimension.\n\n\nIn addition to the high dimensionality, another difficulty in solving the function approximation problem is that the data may be not just high dimensional but unstructured and sparse. A relatively recent idea which has been found very useful in  applications, in fact, too many to    list exhaustively, is to consider the points ${\\bf x}$ as being sampled from an unknown, low dimensional sub--manifold ${\\mathbb X}$ of the ambient high dimensional space ${\\mathbb R}^D$. The understanding of the geometry of ${\\mathbb X}$ is the subject of the bulk of modern research in the area of diffusion geometry. An introduction to this subject can be found in the special issue \\cite{achaspissue} of Applied and Computational Harmonic Analysis. \nThe basic idea is to construct the so--called diffusion matrix from the data, and use its eigen--decomposition for finding local coordinate charts and other useful aspects of the manifold. The convergence of the eigen--decomposition of the matrices to that of the Laplace--Beltrami and other differential operators on the manifold is discussed, for example, in \\cite{belkinfound, lafon, singer}. It is shown in \\cite{jones2008parameter, jones2010universal}  that some of the eigenfunctions on the manifolds yield a local coordinate chart on the manifold. In the context of deep learning, this idea is explored as a function approximation problem in \\cite{cohen_diffusion_net2015}, where a deep net is developed in order to learn the coordinate system given by the eigenfunctions.  \n\nOn the other hand, while much of the research in this direction is focused on understanding the data geometry,  the theoretical foundations for the problems of function approximation and harmonic analysis on such data--defined manifold are developed extensively in \\cite{mauropap, frankbern, modlpmz, eignet, heatkernframe, chuiinterp}. The theory is developed more recently for kernel construction on directed graphs and analysis of functions on changing data in our paper \\cite{tauberian}. However, a drawback of the approach based on data--defined manifolds, known as the out--of--sample extension problem, is that since the diffusion matrix is constructed entirely using the available data, the whole process must be done again if new data become available. A popular idea is then to extend the eigen--functions to the ambient space  by using the so called Nystr\\\"om extension \\cite{coifman2006geometric}. \n\nThe objective of this present paper is to describe a deep learning approach to the problem of function approximation, using three groups of networks in the deep net. The lowest layer accomplishes dimensionality reduction by learning the local coordinate charts on the unknown manifold \\textbf{without using any eigen--decomposition}. Having found the local coordinate system, the problem is reduced to the classical problem of approximating a function on a cube in a relatively low dimensional Euclidean space. For the next two layers, we may now apply the powerful techniques from approximation theory to approximate the target function $f$, given the samples on the training data set ${\\mathcal C}$. We describe two approaches to construct the basis functions using multi--layered neural networks, and to construct other networks to use these basis functions in the next layer to accomplish the desired function approximation. \n\nWe summarize some of the highlights of our paper. \n\\begin{itemize}\n\\item We give a very simple learning method for learning the local coordinate chart near each point. The subsequent approximation process is then entirely local to each coordinate patch.\n\\item Our method allows us to solve the pre--image problem easily; i.e., to generate a point on the manifold  corresponding to a given local coordinate description.\n\\item The learning method itself \\textbf{does not involve} any optimization based technique, except probably for reducing the noise in the values of the function itself.\n\\item We provide optimal error bounds on approximation based on the smoothness of the function, while the method itself \\emph{does not require an a priori knowledge of such smoothness.}\n\\item Our methods can solve easily the \nout--of--sample extension problem. Unlike the Nystr\\\"om extension process, our method does not require any elaborate construction of kernels defined on the ambient space and commuting with certain differential operators on the unknown manifold.\n\\item Our method is designed to control the growth of the out--of--sample extension in a tubular neighborhood of ${\\mathbb X}$, and is local to each coordinate patch.\n\\end{itemize}\n\n This paper is organized as follows. In Section~\\ref{mainsect}, we describe the main ideas in our approach. The local coordinate system is described in detail in Section~\\ref{loccordsect}. Having thus found a local coordinate chart around the input, the problem of function approximation reduces to the classical one. In Section~\\ref{locbasissect}, we demonstrate how the popular basis functions used in this theory can be implemented using neural networks with one or more hidden layers. The function approximation methods which work with unstructured data without using optimization are described in Section~\\ref{approxsect}. In Section~\\ref{extsect}, we explain how our method can be used to solve both the pre--image problem and the out--of--sample extension problem.\n\n\n\n\\bhag{Main ideas and results}\\label{mainsect}\nThe purpose of this paper is to develop a deep learning algorithm to learn a function $f:{\\mathbb X}\\to{\\mathbb R}$, where ${\\mathbb X}$ is a \n$d$ dimensional compact Riemannian sub--manifold  of a Euclidean space ${\\mathbb R}^D$, with $d\\ll D$, given \\emph{training data} of the form $\\{({\\bf x}_j, f({\\bf x}_j))\\}_{j=1}^M$, ${\\bf x}_j\\in{\\mathbb X}$. It is important to note that ${\\mathbb X}$ itself is not known in advance; the points ${\\bf x}_j$ are known only as $D$--dimensional vectors, presumed to lie on ${\\mathbb X}$. In Sub--section~\\ref{ideasect}, we explain our main idea briefly. In Sub--section~\\ref{loccordsect}, we derive a simple construction of the local coordinate chart for ${\\mathbb X}$. In Sub--section~\\ref{locbasissect}, we describe the construction of a neural network with one or more hidden layers to implement two of the basis functions used commonly in function approximation.  While the well known classical approximation algorithms require a specific placement of the training data, one has no control on the location of the data in the current problem. In Section~\\ref{approxsect}, we give algorithms suitable for the purpose of solving this problem.\n\n\n\\subsection{Outline of the main idea}\\label{ideasect}\n\nOur approach  is the following.\n\\begin{enumerate}\n\\item \\label{loccoord} ${\\mathbb X}$ is a finite union of local coordinate neighborhoods, and ${\\bf x}$ belongs to one of them, say $\\mathbb{U}$. We find a local coordinate system for this neighborhood in terms of Euclidean distances on ${\\mathbb R}^D$, say $\\Phi : \\mathbb{U}\\to [-1,1]^d$, where $d$ is the dimension of the manifold. Let ${\\bf y}=\\Phi({\\bf x})$, and with a relabeling for notational convenience, $\\{{\\bf x}_j\\}_{j=1}^K$ be the points in $\\mathbb{U}$, ${\\bf y}_j=\\Phi({\\bf x}_j)$. This way, we have reduced the problem to approximating $g=f\\circ\\Phi :[-1,1]^d\\to{\\mathbb R}$ at ${\\bf y}$, given the values\n$\\{({\\bf y}_j,g({\\bf y}_j))\\}_{j=1}^K$, where $g({\\bf y}_j)=f({\\bf x}_j)$. We note that $\\{{\\bf y}_j\\}$ is a subset of the unit cube of low dimensional Euclidean space, representing a local coordinate patch on ${\\mathbb X}$. Thus, the problem of approximation of $f$ on this patch is reduced that of  approximation of $g$, a well studied classical approximation problem.\n\\item \\label{spline} We will summarize the solution to this problem using neural  networks with one or more hidden layers, e.g., an implementation of multivariate tensor product spline approximation using multi--layerd neural network. \n\\end{enumerate}\nThus, the layers of our deep learning networks will have three main layers.\n\\begin{enumerate}\n\\item The bottom layer receives the input ${\\bf x}$, figures out which of the points ${\\bf x}_j$ are in the coordinate neighborhood of ${\\bf x}$, and computes the local coordinates ${\\bf y}$, ${\\bf y}_j$.\n\\item The next several layers compute the local basis functions necessary for the approximation, for example, the $B$--splines and their translates using the multi--layered neural network as in \\cite{multilayer}.\n\\item The last layer receives the data $\\{({\\bf y}_j,g({\\bf y}_j))\\}_{j=1}^K$, and computes the approximation described in Step~\\ref{spline} above.\n\\end{enumerate}\n\n\\subsection{Local coordinate learning}\\label{loccordsect}\nWe assume that $1\\le d\\le D$ are integers, ${\\mathbb X}$ is a \n$d$ dimensional smooth, compact, connected, Riemannian sub--manifold  of a Euclidean space ${\\mathbb R}^D$, with geodesic distance $\\rho$. \n\nBefore we discuss our own construction of a local coordinate chart on ${\\mathbb X}$, we wish to motivate the work by describing a result from \\cite{jones2010universal}. Let $\\{\\lambda_k^2\\}_{k=0}^\\infty$ be the sequence of eigenvalues of the (negative of  the) Laplace--Beltrami operator on ${\\mathbb X}$, and for each $k\\ge 0$, $\\phi_k$ be the eigenfunction corresponding to the eigenvalue $\\lambda_k^2$.\nWe define a formal ``heat kernel'' by\n\\begin{equation}\\label{heatkerndef}\nK_t({\\bf x},{\\bf y})=\\sum_{k=0}^\\infty \\exp(-\\lambda_k^2t)\\phi_k({\\bf x})\\phi_k({\\bf y}).\n\\end{equation} \nThe following result is a paraphrasing of the heat triangulation theorem proved in \\cite[Theorem~2.2.7]{jones2010universal} under weaker assumptions on ${\\mathbb X}$.\n\n\\begin{theorem}\\label{jmstheo}{\\rm (cf. \\cite[Theorem~2.2.7]{jones2010universal})}\n Let ${\\bf x}_0^*\\in{\\mathbb X}$. There exist constants  $R>0$, $c_1,\\cdots, c_6>0$ depending on ${\\bf x}_0^*$ with the following property. Let $\\mathbf{p}_1,\\cdots,\\mathbf{p}_d$ be $d$ linearly independent vectors in ${\\mathbb R}^d$, and ${\\bf x}_j^*\\in{\\mathbb X}$ be chosen so that ${\\bf x}_j^*-{\\bf x}_0^*$ is in the direction of $\\mathbf{p}_j$, $j=1,\\cdots,d$, and \n$$\nc_1R\\le \\rho({\\bf x}_j^*,{\\bf x}_0^*)\\le c_2R, \\qquad j=1,\\cdots,d,\n$$ \nand $t=c_3R^2$. Let $B\\subset{\\mathbb X}$ be the geodesic ball of radius $c_4R$, centered at ${\\bf x}_0^*$, and \n\\begin{equation}\\label{jmsphidef}\n\\Phi_{\\mbox{jms}}({\\bf x})=R^d(K_t({\\bf x},{\\bf x}_1^*), \\cdots, K_t({\\bf x},{\\bf x}_d^*)), \\qquad {\\bf x}\\in B.\n\\end{equation}\nThen\n\\begin{equation}\\label{jmsdistpreserve}\n\\frac{c_5}{R}\\rho({\\bf x}_1,{\\bf x}_2)\\le \\|\\Phi_{\\mbox{jms}}({\\bf x}_1)-\\Phi_{\\mbox{jms}}({\\bf x}_2)\\|_d\\le \n\\frac{c_6}{R}\\rho({\\bf x}_1,{\\bf x}_2), \\qquad {\\bf x}_1,{\\bf x}_2\\in B.\n\\end{equation}\n\\end{theorem}\nSince the paper \\cite{jones2010universal} deals with a very general manifold, the mapping $\\Phi_{\\mbox{jms}}$ is not claimed to be a diffeomorphism, although it is obviously one--one on $B$.\n\nWe note that even in the simple case of a Euclidean sphere, an explicit expression for the heat kernel is not known. In practice, the heat kernel has to be approximated using appropriate Gaussian networks \\cite{lafon}. In this section, we aim to obtain a local coordinate chart that is computed directly in terms of Euclidean distances on ${\\mathbb R}^D$, and depends upon $d+2$ trainable \nparameters. The construction of this chart constitutes the first hidden layer of our deep learning process. As explained in the introduction, once this chart is in place, the question of function extension on the manifold reduces locally to the well studied problem of function extension on a $d$ dimensional unit cube. \n\nTo describe our constructions,we first develop some notation.\n\nIn this section, it is convenient to use the notation ${\\bf x}=(x^1,\\cdots,x^D)\\in{\\mathbb R}^D$ rather than ${\\bf x}=(x_1,\\cdots,x_D)$, which we will use in the rest of the sections. If $1\\le d\\le D$ is an integer, and ${\\bf x}\\in{\\mathbb R}^d$, $\\|{\\bf x}\\|_d$ denotes the Euclidean norm of ${\\bf x}$. If ${\\bf x}\\in{\\mathbb R}^D$, we will write $\\pi_c({\\bf x})=(x^1,\\cdots,x^d)$, $\\|{\\bf x}\\|_d=\\|\\pi_c({\\bf x})\\|_d$. If ${\\bf x}\\in{\\mathbb R}^d$, $r>0$,\n$$\nB({\\bf x},r)=\\{{\\bf y}\\in{\\mathbb R}^d : \\|{\\bf x}-{\\bf y}\\|_d\\le r\\}.\n$$\n\nThere exists $\\delta^*>0$ with the following properties. The manifold is covered by finitely many geodesic balls such that for the center ${\\bf x}_0^*\\in{\\mathbb X}$ of any of these balls, there exists a diffeomorphism, namely, the exponential coordinate map $u=(u^1,\\cdots,u^D)$ from $B_d(0,\\delta^*)$ to the geodesic ball around ${\\bf x}_0^*=u(0)$ \\cite[p.~65]{docarmo_riemannian}. If $J$ is the Jacobian matrix for $u$, given by $J_{i,j}({\\bf y})=D_iu^j({\\bf y})$, ${\\bf y}\\in B_d(0,\\delta^*)$, then \n\\begin{equation}\\label{finaljacobiatzero}\nJ(0)=[I_d|0_{d,D-d}].\n\\end{equation}\nFurther, there exists $\\kappa>0$ (independent of ${\\bf x}^*$) such that\n\\begin{equation}\\label{jacobimodcont}\n\\|J(\\mathbf{q})-J(0)\\|\\le \\kappa\\|\\mathbf{q}\\|_d, \\qquad \\mathbf{q}\\in B_d(0,\\delta^*).\n\\end{equation}\nLet $\\eta^*:=\\min(\\delta^*, 1\/(2\\kappa))$. Then \\eref{jacobimodcont} implies that \n\\begin{equation}\\label{jacobinormest}\n1\/2\\le 1-\\kappa\\|\\mathbf{q}\\|_d\\le \\|J(\\mathbf{q})\\|\\le 1+\\kappa\\|\\mathbf{q}\\|_d\\le 2, \\qquad \\mathbf{q}\\in B_d(0,\\eta^*).\n\\end{equation}\nIn turn, this leads to\n\\begin{equation}\\label{rhoeuccomp1}\n(1\/2)\\rho(u(\\mathbf{p}),u(\\mathbf{q}))\\le \\|\\mathbf{p}-\\mathbf{q}\\|_d\\le 2\\rho(u(\\mathbf{p}), u(\\mathbf{q})), \\qquad \\mathbf{p},\\mathbf{q}\\in B_d(0,\\eta^*).\n\\end{equation}\n\n\nLet ${\\bf x}_\\ell^*=u(\\mathbf{q}_\\ell)$, $\\ell=1,\\cdots,d$, be chosen with the following properties:\n\\begin{equation}\\label{nbdcond}\n\\|\\mathbf{q}_\\ell\\|_d\\le \\eta^*, \\qquad \\ell=1,\\cdots,d,\n\\end{equation}\nand, with the matrix function $U$ defined by \n\\begin{equation}\\label{umatrixdef}\nU_{i,j}(\\mathbf{q})=u^i(\\mathbf{q})-({\\bf x}_j^*)^i,\n\\end{equation}\n we have\n\\begin{equation}\\label{indepcond}\n\\|J(0)U(0){\\bf y}\\|_d\\ge \\gamma>0, \\qquad \\|{\\bf y}\\|_d= 1.\n\\end{equation}\nAny set $\\{{\\bf x}_\\ell^*\\}$ with these properties will be called  \\emph{coordinate stars} around ${\\bf x}^*$. We note that the matrix $J(0)U(0)$ has columns given by $\\pi_c({\\bf x}^*-{\\bf x}_j^*)$, $j=1,\\cdots,d$, and hence, can be computed without reference to the map $u$. Let \n\\begin{equation}\\label{betastardef}\n\\beta^* := (1\/2)\\min\\left(\\frac{1}{2\\kappa}, \\delta^*,\\frac{\\gamma}{8\\sqrt{d}}\\right).\n\\end{equation} \n\n\\begin{theorem}\\label{loccordtheo}\nLet $\\Psi(\\mathbf{q}):=(\\|u(\\mathbf{q})-u(\\mathbf{q}_\\ell)\\|^2_D)_{\\ell=1}^d\\in{\\mathbb R}^d$. Then \\\\\n{\\rm (a)} $\\Psi$ is a diffeomorphism on $B_d(0,2\\beta^*)$. If $\\mathbf{p}, \\mathbf{q}\\in B_d(0,2\\beta^*)$, ${\\bf x}=u(\\mathbf{p})$, ${\\bf y}=u(\\mathbf{q})$, then\n\\begin{equation}\\label{distortest}\n\\frac{\\gamma}{2}\\rho({\\bf x},{\\bf y})\\le \\|\\Psi(\\mathbf{p})-\\Psi(\\mathbf{q})\\|_d \\le 32\\sqrt{d}\\eta^*\\rho({\\bf x},{\\bf y}).\n\\end{equation}\n{\\rm (b)} The function $\\Psi$ is a diffeomorphism from $B_d(0,\\beta^*)$ onto $B_d(\\Psi(0),\\beta^*)$. \n \\end{theorem}\n \n\\begin{rem}\\label{maptocuberem}\n{\\rm\nLet $\\mathbb{B}=u(B_d(0,\\beta^*))\\subset {\\mathbb X}$ be a geodesic ball around ${\\bf x}_0^*$. For ${\\bf x}\\in \\mathbb{B}$, we define\n$$\n\\phi({\\bf x})=\\Psi(u^{-1}({\\bf x}))=(\\|{\\bf x}-{\\bf x}_\\ell^*\\|_D^2).\n$$\nThen Theorem~\\ref{loccordtheo}(b) shows that \n$\\phi$ is a diffeomorphism from $\\mathbb{B}$ onto $B_d(\\Psi(0),\\beta^*)$. Since $\\Psi(0)=(\\|{\\bf x}_0^*-{\\bf x}_\\ell^*\\|_D^2)$, \n$$\n\\Phi({\\bf x})=\\frac{\\sqrt{d}}{\\beta^*}(\\phi({\\bf x})-\\Psi(0)), \\qquad {\\bf x}\\in \\mathbb{B}\n$$\nmaps $\\mathbb{B}$ diffeomorphically onto $B_d(0,\\sqrt{d})\\supset [-1,1]^d$. Let $\\mathbb{U}=\\Phi^{-1}([-1,1]^d)$. Then  $\\mathbb{U}$ is a neighborhood of ${\\bf x}_0^*$ and $\\Phi$ maps $\\mathbb{U}$ diffeomorphically onto $[-1,1]^d$. We oberve that ${\\mathbb X}$ is a union of finitely many neighborhoods of the form $\\mathbb{U}$, so that any ${\\bf x}\\in{\\mathbb X}$ belongs to at least one such neighborhood. Moreover, $\\Phi({\\bf x})$ can be computed entirely in terms of the description of ${\\bf x}$ in terms of its $D$--dimensional coordinates. \\hfill$\\Box$\\par\\medskip\n}\n\\end{rem}\n\n\\begin{rem}\\label{trainrem}\n{\\rm The trainable parameters are thus $\\beta^*$, and the points ${\\bf x}_0^*, \\cdots, {\\bf x}_d^*$. Since $\\|J(0)\\|=1$, the condition \\eref{indepcond} is satisfied if ${\\bf x}_\\ell^*-{\\bf x}_0^*$ are along linearly independent directions as in Theorem~\\ref{jmstheo}.\n\\hfill$\\Box$\\par\\medskip}\n\\end{rem}\n\n\\begin{rem}\\label{networkimprem}\n{\\rm\nSince the mapping $\\Phi$ in Remark~\\ref{maptocuberem} is a quadratic polynomial in ${\\bf x}$, it can be implemented as a neural network with a single hidden layer using the activation function given in \\eref{requdef} as described in Sub--Section~\\ref{polysubsect}.\n}\n\\end{rem}\n\n\\begin{uda}\\label{helixexample}\n{\\rm\nLet $0<a<1$, and $M=M_a$ be the circular helix defined by\n$$\n\\mathbf{u}(s)=(\\cos as, \\sin as, \\sqrt{1-a^2}s)^T.\n$$\nClearly, $M$ is a one dimensional manifold, and $s$ is the arclength parameter, measured from $(1,0,0)^T$. The curvature at any point is $a^2$. For any point ${\\bf z}_0\\in M$, $U_{{\\bf z}_0}=M$, with the diffeomorphism given by $\\mathbf{u}(\\tan^{-1}(\\pi t\/2))$, $t\\in (-1,1)$. An interesting fact is that $\\|\\mathbf{u}(s+2\\pi)-\\mathbf{u}(s)\\|_3=2\\pi\\sqrt{1-a^2}$ can be made arbitrarily small by choosing $a$ close to $1$, even though the geodesic distance between $\\mathbf{u}(s+2\\pi)$ and $\\mathbf{u}(s)$ is $2\\pi$. Let $s_0\\in{\\mathbb R}$, $s_0+\\pi\/4\\le s_1\\le s_0+3\\pi\/8$, and $U := \\{\\mathbf{u}(s)\\ :\\ |s-s_0|\\le \\pi\/8\\}$. Let $\\Psi(s):=\\|\\mathbf{u}(s)-\\mathbf{u}(s_1)\\|_3^2$. It is easy to calculate that\n$$\n\\Psi(s)= 2-2\\cos (a(s-s_1))+(1-a^2)(s-s_1)^2,\n$$\nso that\n$$\n\\Psi'(s)=2a\\sin (a(s-s_1))+ 2(1-a^2)(s-s_1).\n$$\nIf $\\mathbf{u}(s)\\in U$, then $\\pi\/8\\le s_1-s\\le \\pi\/2$. Therefore, using the well known estimates\n$$\n\\frac{2\\theta}{\\pi} \\le \\sin (\\theta) \\le \\theta, \\qquad \\theta\\in [0,\\pi\/2],\n$$\nwe obtain that\n\\begin{equation}\\label{snakeupbd}\n|\\Psi'(s)| \\le 2a^2(s_1-s)+2(1-a^2)(s_1-s)\\le \\pi,\n\\end{equation}\nand\n\\begin{equation}\\label{snakelowbd}\n|\\Psi'(s)| \\ge \\frac{4a^2}{\\pi}(s_1-s) + 2(1-a^2)(s_1-s)\\ge \\frac{\\pi}{8}(2-2a^2(1-2\/\\pi))\\ge 1\/2.\n\\end{equation}\nHence, for any points $\\mathbf{u}(t_1), \\mathbf{u}(t_2)\\in U$, we have\n$$\n(1\/2)|t_1-t_2|\\le \\left|\\|\\mathbf{u}(t_1)-\\mathbf{u}(s_1)\\|_3^2-\\|\\mathbf{u}(t_2)-\\mathbf{u}(s_1)\\|_3^2\\right|\\le \\pi |t_1-t_2|.\n$$\nWe note that  the neighborhood $U$ where this estimate holds and the constants are independent of the curvature.\n\\hfill$\\Box$\\par\\medskip\n}\n\\end{uda}\nThe remainder of this section is devoted to the proof of Theorem~\\ref{loccordtheo}.\n\n\\begin{lemma}\\label{umatrixlemma}\n Let $\\mathbf{q} \\in B_d(0,\\eta^*)$.  Each of the following statements hold for the matrix $U$ defined in \\eref{umatrixdef}:\n\\begin{equation}\\label{ucontinuity}\n \\|U(\\mathbf{q})-U(0)\\| \\le 2\\sqrt{d}\\|\\mathbf{q}\\|_d,\n\\end{equation}\n\\begin{equation}\\label{uupest}\n\\|U(\\mathbf{q})\\|\\le 2\\sqrt{d}\\max_{1\\le\\ell\\le d}\\|\\mathbf{q}-\\mathbf{q}_\\ell\\|_d\\le 4\\sqrt{d}\\eta^*,\n\\end{equation}\n\\begin{equation}\\label{scriptjcont}\n\\|J(\\mathbf{q})U(\\mathbf{q})-J(0)U(0)\\| \\le 4\\sqrt{d}\\|\\mathbf{q}\\|_d.\n\\end{equation}\nWith $\\beta^*$ as in \\eref{betastardef}, for $\\|\\mathbf{q}\\|_d\\le 2\\beta^*$, $\\|{\\bf y}\\|_d=1$,\n\\begin{equation}\\label{ulowest}\n\\|J(\\mathbf{q})U(\\mathbf{q}){\\bf y}\\|_d\\ge  \\gamma\/2.\n\\end{equation}\n\\end{lemma}\n\n\\begin{Proof}\\\nIn view of \\eref{jacobinormest} and the mean value theorem, we have for $\\|\\mathbf{p}\\|_d\\le \\eta^*$,\n\\begin{equation}\\label{pf1eqn2}\n \\|u(\\mathbf{q})-u(\\mathbf{p})\\|_D \\le 2\\|\\mathbf{q}-\\mathbf{p}\\|_d.\n\\end{equation}\n\nWe observe further that for any integers $m, \\ell$, $U(\\mathbf{q})_{m,\\ell}-U(0)_{m,\\ell}= u^m(\\mathbf{q})-u^m(0)$. Consequently, for any ${\\bf y}\\in{\\mathbb R}^d$, $\\|{\\bf y}\\|_d\\le 1$,\n$$\n\\|(U(\\mathbf{q})-U(0)){\\bf y}\\|_D=\\|u(\\mathbf{q})-u(0)\\|_D\\sum_{\\ell=1}^d |y^\\ell|\\le 2\\sqrt{d}\\|\\mathbf{q}\\|_d\\|{\\bf y}\\|_d.\n$$\nThis proves \\eref{ucontinuity}.\n\nIn view of \\eref{pf1eqn2}, used with $\\mathbf{q}_\\ell$ in place of $\\mathbf{p}$, $\\ell=1,\\cdots,d$, we obtain for all ${\\bf y}\\in{\\mathbb R}^d$, $\\|{\\bf y}\\|_d\\le 1$,\n$$\n\\left\\|\\sum_{\\ell=1}^d {\\bf y}^\\ell (u(\\mathbf{q})-u(\\mathbf{q}_\\ell) )\\right\\|_D \\le \\sum_{\\ell=1}^d |{\\bf y}^\\ell|\\|u(\\mathbf{q})-u(\\mathbf{q}_\\ell)\\|_D \\le 2\\sqrt{d}\\max_{1\\le \\ell\\le d}\\|\\mathbf{q}-\\mathbf{q}_\\ell\\|_d.\n$$\nThis proves \\eref{uupest}.\n\nIn view of \\eref{jacobimodcont}, \\eref{uupest}, \\eref{ucontinuity}, we obtain for $\\|\\mathbf{q}\\|_d\\le \\eta^*\\le 1\/(2\\kappa)$ that \n\\begin{eqnarray*}\n\\|J(\\mathbf{q})U(\\mathbf{q})-J(0)U(0)\\|&=&\\|(J(\\mathbf{q})-J(0))U(\\mathbf{q}) + J(0)(U(\\mathbf{q})-U(0))\\|\\\\\n&\\le& \\|J(\\mathbf{q})-J(0)\\|\\|U(\\mathbf{q})\\| +\\|J(0)\\|\\|U(\\mathbf{q})-U(0)\\| \\\\\n&\\le& 4\\sqrt{d}\\eta^*\\kappa\\|\\mathbf{q}\\|_d + 2\\sqrt{d}\\|\\mathbf{q}\\|_d= 2\\sqrt{d}(1+2\\eta^*\\kappa)\\|\\mathbf{q}\\|_d\\le 4\\sqrt{d}\\|\\mathbf{q}\\|_d.\n\\end{eqnarray*}\nThis proves \\eref{scriptjcont}. The estimate \\eref{ulowest} follows easily from this and \\eref{indepcond}.\n\\end{Proof}\n\n\\noindent\n\\textsc{Proof of Theorem~\\ref{loccordtheo}.}\nIn this proof only, let  ${\\cal J}(\\mathbf{q})$ be the Jacobian of $\\Psi$: ${\\cal J}_{i,j}(\\mathbf{q})=D_i(\\|u(\\mathbf{q})-u(\\mathbf{q}_j)\\|^2)$. Then ${\\cal J}(\\mathbf{q})=2J(\\mathbf{q})U(\\mathbf{q})$. The estimate \\eref{indepcond} shows that ${\\cal J}(0)$ is invertible, and that $\\|{\\cal J}(0)^{-1}\\|\\le 1\/(2\\gamma)$. The estimate \\eref{scriptjcont} then shows that\n\\begin{equation}\\label{pf2eqn1}\n\\|{\\cal J}(\\mathbf{q})-{\\cal J}(0)\\| \\le 1\/(2\\|{\\cal J}(0)^{-1}\\|), \\qquad \\|\\mathbf{q}\\|_d\\le 2\\beta^*.\n\\end{equation}\nTherefore, the inverse function theorem as given in \\cite[Theorem~9.24 and its proof]{rudin1976principles} implies that $\\Psi$ is a diffeomorphism on $B(0,2\\beta^*)$ as claimed. For $\\|\\mathbf{q}\\|_d\\le 2\\beta^*$, \\eref{ulowest} shows that $\\|{\\cal J}(\\mathbf{q})^{-1}\\|\\le  1\/\\gamma$. Also, \\eref{uupest} and \\eref{jacobinormest}  show that $\\|{\\cal J}(\\mathbf{q})\\|\\le 16\\sqrt{d}\\eta^*$. Hence, the mean value theorem implies that\n\\begin{equation}\\label{pf2eqn2}\n\\gamma \\|\\mathbf{q}-\\mathbf{p}\\|_d\\le \\|\\Psi(\\mathbf{q})-\\Psi(\\mathbf{p})\\|_d \\le 16\\sqrt{d}\\eta^* \\|\\mathbf{q}-\\mathbf{p}\\|_d.\n\\end{equation}\nTogether with \\eref{rhoeuccomp1}, this implies \\eref{distortest}.\n\nThe part (b) follows also from \\cite[Theorem~9.24 and its proof]{rudin1976principles} and \\eref{pf2eqn1}. \n\\hfill$\\Box$\\par\\medskip\n\n\n \\subsection{Local basis functions}\\label{locbasissect}\nHaving found a local coordinate map $\\Phi$ on a neighborhood $\\mathbb{U}$ of ${\\bf x}$ on ${\\mathbb X}$, the problem of extending $f$ from $\\{{\\bf x}_j\\}\\cap \\mathbb{U}$ to $\\mathbb{U}$ is reduced to extending $f\\circ\\Phi$ from ${\\mathcal C}=\\{{\\bf y}_j=\\Phi({\\bf x}_j)\\}\\subset [-1,1]^d$, a classical approximation problem. There is, of course, 100+ years of research on this subject. We restrict ourselves to two examples, which can be implemented using neural networks with one or more hidden layers.  One of the most popular activation function in the deep learning literature \n(e.g., \\cite{lecun2015deep}) is the \nrectified linear unit function\n$$\nt_+=\\max(0,t).\n$$\nSince this function is not continuously differentiable, there are some technical difficulties to use common algorithms like back--propagation with these activation functions. Although we do not need back--propagation in our theory, we prefer to deal with a \\emph{rectified quadratic unit function} defined for $t\\in{\\mathbb R}$ by\n\\begin{equation}\\label{requdef}\n\\sigma(t)=\\left\\{\\begin{array}{ll}\nt^2, &\\mbox{ if $t\\ge 0$,}\\\\\n0, &\\mbox{ if $t<0$,}\n\\end{array}\\right.\n\\end{equation}\nwhich is continuously differentiable on ${\\mathbb R}$. Our theory will work in general with any activation function  of order $k\\ge 2$; i.e., with a function $\\sigma$ that satisfies\n\\begin{equation}\\label{actfnorder}\n\\lim_{t\\to -\\infty}\\frac{\\sigma(t)}{t^k}=0, \\qquad \\lim_{t\\to \\infty}\\frac{\\sigma(t)}{t^k}=1,\n\\end{equation}\nbut for the sake of clarity of exposition, we will use only the activation function $\\sigma$ defined in \\eref{requdef}.\n\n\\subsection{Polynomials}\\label{polysubsect}\nThe most basic class of classical approximants is the set of all polynomials. For $n>0$, we denote the class of all algebraic polynomials of coordinatewise degree at most $n$ in $d$ variables by $\\Pi_n^d$. (It is convenient to use the same notation also when $n$ is not an integer; in this case, $\\Pi_n^d$ is just $\\Pi_{\\lfloor n\\rfloor}^d$.\n\nThe basic implementation of polynomials is given in \\cite[Proof of Theorem~3.1]{chuili1992}, where an explicit construction is given for finding the weights $\\{{\\bf w}_k\\}$, the thresholds $\\{b_k\\}$ and the coefficients $\\{a_k\\}$ used in \\eref{chuilipolynet} below.\n\n\\begin{theorem}\\label{chuilitheo}\nLet $n>0$,  $N=2^{\\lceil \\log_2 n\\rceil}$,\n$P\\in\\Pi_N^d\\supseteq \\Pi_n^d$,\nthen there exist weights ${\\bf w}_1,\\cdots,w_{\\mbox{dim}(\\Pi_N^d)}$ and real numbers $a_1,\\cdots, a_{\\mbox{dim}(\\Pi_N^d)}$, $b_1,\\cdots, b_{\\mbox{dim}(\\Pi_N^d)}$ such that\n\\begin{equation}\\label{chuilipolynet}\nP({\\bf x})=\\sum_{k=1}^{\\mbox{dim}(\\Pi_N^d)}a_k({\\bf w}_k\\cdot{\\bf x} +b_k)^N, \\qquad x\\in {\\mathbb R}^d.\n\\end{equation}\nHere, the weights $\\{{\\bf w}_k\\}$ and the thresholds $\\{b_k\\}$ are independent of $P$ and the coefficients $\\{a_k\\}$ are linear functionals on $\\Pi_N^d$.\n\\end{theorem}\n\nWe observe that\n\\begin{equation}\\label{powerfromtruncpow}\nt^2=\\sigma(t)+\\sigma(-t), \\qquad x\\in{\\mathbb R},\n\\end{equation}\nwhile\n\\begin{equation}\\label{iterpower}\nt^N=((t^2)^2\\cdots)^2, (\\log_2 N \\mbox{ times}),\n\\end{equation}\nso that \nthe expression on the right hand side of \\eref{chuilipolynet} can be expressed as a neural network with $\\log_2 N$ hidden layers. \n\nWe note that a neural network with one hidden layer  is given in \\cite{optneur}, but using a $C^\\infty$ activation function $\\sigma$; e.g.,\n$\\sigma(t)= (1+e^{-t})^{-1}$. This uses the fact that for ${\\bf w},{\\bf x}\\in{\\mathbb R}^d$ and $b\\in{\\mathbb R}$, such that none of the derivatives $\\sigma^{(j)}(b)$, $j=0,1,\\cdots$, equal to $0$,\n\\begin{equation}\\label{optneurformula}\n\\frac{\\partial^{|\\mathbf{k}|}}{\\partial \\mathbf{w}^{\\mathbf{k}}}\\sigma(\\mathbf{w}\\cdot{\\bf x}+b)\\bigg|_{\\mathbf{w}=0} ={\\bf x}^{\\mathbf{k}}\\sigma^{|\\mathbf{k}|}(b).\n\\end{equation}\nA finite difference scheme to implement this differentiation yields a neural network with one hidden layer, containing exactly $\\mbox{dim}(\\Pi_n^d)$ neurons, and should be stable for $C^\\infty$ functions. If stability is a greater concern, then one may use other  numerical differentiation schemes to implement this formula, e.g., spectral methods \\cite{gottlieb1977numerical}. \n\n\\subsection{$B$--splines}\\label{splinesubsect}\n\nFor $t\\in{\\mathbb R}$, and integer $m\\ge 1$, let \n$$\nt^m_+=\\left\\{\\begin{array}{ll}\nt^m, &\\mbox{ if $t\\ge 0$,}\\\\\n0, &\\mbox{otherwise.}\n\\end{array}\\right.\n$$\nA tensor product cardinal $B$-spline at ${\\bf y}\\in [-1,1]^d$ is defined by\n\\begin{equation}\\label{tensbsplinedef}\nN_m({\\bf y})=\\frac{1}{(m-1)!^d}\\sum_{{\\bf k}\\in{\\mathbb Z}, \\atop {\\bf k}\\ge 0,\\ |{\\bf k}|_\\infty\\le m} (-1)^{|{\\bf k}|_1}\\prod_{j=1}^d {m\\choose k_j}\\prod_{j=1}^d (y_j-k_j)^m_+.\n\\end{equation}\n\n\nIt is explained in \\cite{multilayer,mhaskar1993neural} that the quantity $N_m({\\bf y})$ can be computed using a neural network with a sigmoidal function of order $m-1$ consisting of finitely many neurons arranged in multiple hidden layers (the number of neurons and layers depending on $m$ and $d$ alone). \nThus, if $m$ is a power of $2$, then each of the terms $(y_j-k_j)^m_+$ can be implemented as an iterated power of $(y_j-k_j)^2_+$ (cf. \\eref{iterpower}.) The product of $d$ such expressions can be implemented using either Theorem~\\ref{chuilitheo} as a network with mulitple hidden layer and utilizing the rectified quadratic unit function as the activation function, or a discretization of the formula \\eref{optneurformula} using a $C^\\infty$ sigmoidal function as explained in Sub--section~\\ref{polysubsect}.\n\n\n\n\n\n\n\\subsection{Function approximation}\\label{approxsect}\nIn this section, if ${\\bf y}\\in{\\mathbb R}^d$, $\\|{\\bf y}\\|_\\infty$ is the $\\ell^\\infty$ norm of ${\\bf y}$.\n\\subsubsection{Spline based approximation}\\label{splineapproxsect}\n\nIn \\cite[Section~4.5]{chuiwaveletbk},  \\cite{chui_diamond_quasi_int90}, a quasi--interpolatory spline function is defined by\n\\begin{equation}\\label{quasisplinedef}\nQ_m(f)({\\bf y})=\\sum_{{\\bf k}\\in{\\mathbb Z}^d}\\lambda_m^*(f(\\cdot+{\\bf k}))N_m({\\bf y}+m\/2+{\\bf k}),\n\\qquad {\\bf y}\\in{\\mathbb R}^d,\n\\end{equation}\nwhere $\\lambda_m^*$ are compactly supported linear functionals, designed specifically to ensure that $Q_m(P)=P$ for every polynomial $P$ of coordinatewise degree at most $m-1$ in $d$ variables. With $Q_{m,h}(f)({\\bf y})=Q_m(f(h(\\cdot)))({\\bf y}\/h)$, $h>0$, one has the approximation bound for small $h$:\n\\begin{equation}\\label{quasi_spline_approx_bd}\n\\max_{{\\bf y}\\in [-1,1]^d}|Q_{m,h}(f)({\\bf y})-f({\\bf y})|={\\cal O}(h^m).\n\\end{equation}\nThe linear functionals $\\lambda_m^*$ are based on \\emph{finitely many} samples of $f$ at the grid points in a compact subset of ${\\mathbb Z}^d$. In our context, the data for approximating $f$ is not in this form. Therefore, we may use the following algorithm given in \\cite{quasiint2000}, where we assume that $\\lambda_m^*$ is scaled so as to be supported on $[-1,1]^d$.\n\\begin{quotation}\n\\noindent {\\bf Given:} A  set ${\\mathcal C}=\\{\\bs{\\xi}_j\\}$ of points in $[-1,1]^d$. Let\n$$\n\\delta({\\mathcal C})=\\max_{\\bs{\\xi}_1-\\bs{\\xi}_2\\in{\\mathcal C}}|\\bs{\\xi}_1-\\bs{\\xi}_2|_\\infty,\n$$\nand $\\delta({\\mathcal C})$ be sufficiently small.\\\\\n\n\\noindent {\\bf Objective:} To find real numbers\n$\\{a_{\\bs{\\xi}}\\}_{\\bs{\\xi}\\in{\\mathcal C}}$ such that the functional\n\\begin{equation}\\label{splinequad}\n\\gamma(f) := \\sum_{\\bs{\\xi}\\in{\\mathcal C}} a_{\\bs{\\xi}} f(\\bs{\\xi})\n\\end{equation}\nsatisfies\n$$\n\\gamma(P) = \\lambda_m^*(P), \\qquad \\mbox{ if $P$ is $d$--variate polynomial of coordinatewise degree $\\le m-1$.}\n$$\n{\\bf Steps:}\n\\begin{enumerate}\n\\item Divide $[-1,1]^d$ into congruent subcubes of side not exceeding $2\\delta({\\mathcal C})$. \n\\item Choose  ${\\mathcal C}_0\\subseteq {\\mathcal C}$, so that each subcube has exactly one point of ${\\mathcal C}_0$.\n\\item Solve the following (underdetermined) system of equations for\nthe unknowns $a_{\\bs{\\xi}}$, $\\bs{\\xi}\\in{\\mathcal C}_0$.\n\n\\begin{equation}\\label{axieqn}\n\\sum_{\\bs{\\xi}\\in{\\mathcal C}}a_{\\bs{\\xi}} \\bs{\\xi}^{\\bf k} = \\lambda_m^*((\\cdot)^{\\bf k}), \\qquad |{\\bf k}|_\\infty\\le m-1.\n\\end{equation}\n\n\\item Set $a_{\\bs{\\xi}}:=0$ if $\\bs{\\xi}\\in{\\mathcal C}\\setminus {\\mathcal C}_0$.\n\\item Output $(a_{\\bs{\\xi}})_{\\bs{\\xi}\\in{\\mathcal C}}$.\n\\end{enumerate}\n\\end{quotation}\nSubstituting $\\gamma$ in place of $\\lambda_m^*$ in the definition of $Q_m(f)$ yields the desired spline approximation\n\\begin{equation}\\label{scat_quasi_spline_def}\n\\tilde{Q}_m(f)({\\bf y})=\\sum_{{\\bf k}\\in{\\mathbb Z}^d}\\gamma(f(\\cdot+{\\bf k}))N_m({\\bf y}+m\/2+{\\bf k}),\n\\qquad {\\bf y}\\in{\\mathbb R}^d,\n\\end{equation}\nand it is proved in \\cite{quasiint2000} that the estimate \\eref{quasi_spline_approx_bd} holds with $\\tilde{Q}_m$ replacing $Q_m$. \n\n\\subsubsection{Polynomial quasi--interpolation}\\label{polyapproxsect}\n\nA standard method for polynomial approximation is to consider a filtered projection defined in \\eref{summopdef} below. \n\nThe Chebyshev polynomials (of first kind) are defined recursively for $t\\in{\\mathbb R}$ and integer $m\\ge 0$ by\n\\begin{equation}\\label{chebpolydef}\nT_0(t)=1,\\quad T_1(t)=t,\\quad T_m(t)=2tT_{m-1}(t)-T_{m-2}(t).\n\\end{equation}\nIn terms of monomials, the Chebyshev polynomials are given by\n\\begin{equation}\\label{chebmonoexp}\nT_{2n}(t)=\\sum_{j=0}^n \\frac{(-4)^j}{(2j)!}\\prod_{\\ell=1}^j (n^2-(j-\\ell)^2))t^{2j},\\quad T_{2n+1}(t)=\\sum_{j=0}^n \\frac{(-1)^j(2n+1)^2}{(2j+1)!}\\prod_{\\ell=1}^j ((2n+1)^2-(2j-2\\ell+1)^2)t^{2j+1}.\n\\end{equation}\nFor ${\\bf y}\\in{\\mathbb R}^d$ and multi--integer $\\mathbf{m}=(m^1,\\cdots,m^d)\\ge 0$, the tensor product Chebyshev polynomial is defined by\n\\begin{equation}\\label{tenschebpolydef}\nT_{\\mathbf{m}}({\\bf y})=\\prod_{j=1}^d T_{m^j}(y^j).\n\\end{equation}\n\nWe choose a smooth low pass filter $h$; i.e., an even function $h :{\\mathbb R}\\to [0,1]$ such that $h(u)=1$ if $|u|\\le 1\/2$, and $h(u)=0$ if $|u|\\ge 1$, and abuse the notation as usual to define\n$$\nh(\\mathbf{u})=\\prod_{j=1}^d h(u^j).\n$$\nWith this filter, we define the kernel\n\\begin{equation}\\label{kerndef}\n\\Phi_n({\\bf y},\\mathbf{t})=\\sum_{{\\bf k}\\in {\\mathbb Z}^d} h({\\bf k}\/n)T_{\\bf k}({\\bf y})T_{\\bf k}(\\mathbf{t}), \\qquad n>0, \\ {\\bf y},\\mathbf{t}\\in [-1,1]^d.\n\\end{equation}\nThen the filtered projection operator is defined by\n\\begin{equation}\\label{summopdef}\nV_n(f)({\\bf y})=\\int_{[-1,1]^d} f(\\mathbf{t})\\Phi_n({\\bf y}, \\mathbf{t})\\frac{d\\mathbf{t}}{\\sqrt{(1-(t^1)^2)\\cdots (1-(t^d)^2)}}, \\qquad {\\bf y}\\in [-1,1]^d,\n\\end{equation}\nIt is well known that if $f$ is any continuous function on $[-1,1]^d$, then $\\{V_n(f)\\}$ converges uniformly to $f$ at the near optimal rate of approximation. For example, if $f$ has partial derivatives up to order $r$ in each variable, then analogously to \\eref{quasi_spline_approx_bd}, but for large $n$ rather than small $h$,\n\\begin{equation}\\label{goodapproxbd}\n\\max_{{\\bf y}\\in [-1,1]^d}|V_n(f)({\\bf y})-f({\\bf y})| ={\\cal O}(n^{-r}).\n\\end{equation}\nTheoretically, the question then is to compute $V_n(f)$ using the data ${\\mathcal C}$ as in Sub--section~\\ref{splineapproxsect}. The procedure we describe below from \\cite{indiapap, jaenpap, mnpdiabetes} also describes the choice of the parameter $n$ depending upon the data. \n\\begin{quotation}\n\\noindent {\\bf Given:} A  set ${\\mathcal C}=\\{\\bs{\\xi}_j\\}$ of points in $[-1,1]^d$. Let\n$$\n\\delta^\\circ({\\mathcal C})=\\max_{\\bs{\\xi}_1,\\bs{\\xi}_2\\in{\\mathcal C}}\\max_{1\\le j\\le d}|\\arccos(\\xi^j_1)-\\arccos({\\bf x}^j_2)|.\n$$\nand $\\delta^\\circ({\\mathcal C})$ be sufficiently small. We also fix an integer $n>0$.\\\\\n\n\\noindent {\\bf Objective:} To find real numbers\n$\\{w_{\\bs{\\xi}}\\}_{\\bs{\\xi}\\in{\\mathcal C}}$ such that the functional\n\\begin{equation}\\label{polyquad}\n\\gamma^\\circ(f) := \\sum_{\\bs{\\xi}\\in{\\mathcal C}} w_{\\bs{\\xi}} f(\\bs{\\xi})\n\\end{equation}\nsatisfies\n$$\n\\gamma^\\circ(P) = \\int_{[-1,1]^d}\\frac{P(\\mathbf{t})d\\mathbf{t}}{\\sqrt{(1-(t^1)^2)\\cdots (1-(t^d)^2)}}, \n$$\nfor all  $d$--variate polynomials $P$ of coordinatewise degree $\\le n-1$.\\\\\n\n\\noindent{\\bf Steps:}\n\\begin{enumerate}\n\\item Divide $[-1,1]^d$ into congruent subcubes of side not exceeding $2\\delta^\\circ({\\mathcal C})$. \n\\item Choose  ${\\mathcal C}_0\\subseteq {\\mathcal C}$, so that each subcube has exactly one point of ${\\mathcal C}_0$.\n\\item Solve the following (underdetermined) system of equations for\nthe unknowns $w_{\\bs{\\xi}}$, $\\bs{\\xi}\\in{\\mathcal C}_0$.\n\n\\begin{equation}\\label{wxieqn}\n\\sum_{\\bs{\\xi}\\in{\\mathcal C}}w_{\\bs{\\xi}} T_{\\bf k}(\\bs{\\xi}) = \\int_{[-1,1]^d}\\frac{T_{\\bf k}(\\mathbf{t})d\\mathbf{t}}{\\sqrt{(1-(t^1)^2)\\cdots (1-(t^d)^2)}}=\\delta_{{\\bf k},0}, \\qquad |{\\bf k}|_\\infty\\le n-1.\n\\end{equation}\n\n\\item Set $w_{\\bs{\\xi}}:=0$ if $\\bs{\\xi}\\in{\\mathcal C}\\setminus {\\mathcal C}_0$.\n\\item Output $(w_{\\bs{\\xi}})_{\\bs{\\xi}\\in{\\mathcal C}}$.\n\\end{enumerate}\n\\end{quotation}\nIt is proved in the papers cited above that with the discretized operator\n\\begin{equation}\\label{discsummopdef}\nV_{n,{\\mathcal C}}(f)({\\bf y})=\\sum_{\\bs{\\xi}\\in{\\mathcal C}}w_{\\bs{\\xi}}f(\\bs{\\xi})\\Phi_n({\\bf y},\\bs{\\xi}),\n\\end{equation}\none obtains the near best rates of approximation. In particular, if $f$ has partial derivatives up to order $r$ in each variable, then \\eref{goodapproxbd} holds with $V_{n,{\\mathcal C}}(f)$ replacing $V_n(f)$. In practice, we choose $n$ to be the largest integer such that either the condition number of the system of equations in \\eref{wxieqn} is ``reasonable'' or else by checking the resulting errors in \\eref{wxieqn} \\cite{quadconst}.\n\nThe formula \\eref{discsummopdef} can be re--written in the form\n\\begin{equation}\\label{disc_summ_four}\nV_{n,{\\mathcal C}}(f)({\\bf y})=\\sum_{|{\\bf k}|_\\infty\\le n-1} h({\\bf k}\/n)\\hat{f}({\\mathcal C},{\\bf k})T_{\\bf k}({\\bf y}),\n\\end{equation}\nwhere\n\\begin{equation}\\label{disc_four_coeff}\n\\hat{f}({\\mathcal C},{\\bf k})=\\sum_{\\bs{\\xi}\\in {\\mathcal C}} w_{\\bs{\\xi}}f(\\bs{\\xi})T_{\\bf k}(\\bs{\\xi}).\n\\end{equation}\nTherefore, rather than evaluating the Chebyshev polynomials as defined in \\eref{tenschebpolydef}, \\eref{chebmonoexp}, one can use a \nmulti--layered network to evaluate $V_{n,{\\mathcal C}}(f)$ in a more stable manner as follows. The first layer computes the coefficients $h({\\bf k}\/n)\\hat{f}({\\mathcal C},{\\bf k})$ using the available data. The output of this layer is input to a recurrent network to execute a multi--variate version of the well known  Clenshaw algorithm \\cite[pp.~78--80]{gautschibk}.\n\n\\bhag{Extensions}\\label{extsect}\n\nSince the starting point of diffusion geometry is to consider eigen--decomposition of a diffusion matrix, which is constructed using the available data, the entire computation needs to be redone if a new data becomes available. Since the manifold ${\\mathbb X}$ is only an abstract model, it is not even clear that the new data will belong to this manifold. This gives rise to two related questions. One is to find new points on the manifold; the so called pre--image problem \\cite{cohen_diffusion_net2015} and the other is the out-of-sample extension problem; i.e., extend the target function to points not necessarily on ${\\mathbb X}$. In this section, we make some comments on how to use the theory described in the previous sections for solving these problems. \n\n\\subsection{Pre--image problem}\\label{preimagesect}\n\n In Remark~\\ref{maptocuberem}, we have given an onto diffeomorphism $\\Phi : \\mathbb{U} \\to [-1,1]^d$ where  $\\mathbb{U}\\subset {\\mathbb X}$.  The pre--image problem is the following. Given a point ${\\bf y}\\in [-1,1]^d$, find ${\\bf x}\\in{\\mathbb X}$ such that $\\Phi({\\bf x})={\\bf y}$.  This amounts to approximating the $D$--output function $\\Phi^{-1}$ on $[-1,1]^d$, given its values at the known points $\\{\\Phi(\\bs{\\xi}) : \\bs{\\xi}\\in {\\mathcal C}\\}$, and can therefore be solved using any of the techniques described in the previous sections.\n \n\\subsection{Out of sample extension}\\label{oosesect}\n\nOne well known strategy for function extension outside the manifold is the following. One starts with a compact, positive--semi--definite symmetric kernel $K :{\\mathbb R}^D\\times {\\mathbb R}^D$ and considers the eigen--decomposition of $K$ restricted to ${\\mathbb X}\\times {\\mathbb X}$; thus, for example, if $\\mu^*$ is the volume measure on ${\\mathbb X}$, one finds numbers $\\lambda_k\\ge 0$ and orthonormal functions $\\phi_k$ on ${\\mathbb X}$ such that\n\\begin{equation}\\label{kerneleigdef}\n\\int_{\\mathbb X} K({\\bf x},{\\bf y})\\phi_k({\\bf y})d\\mu^*({\\bf y})=\\lambda_k\\phi_k({\\bf x}), \\qquad k=0,1,\\cdots, \\ {\\bf x}\\in{\\mathbb X}.\n\\end{equation}\nA function on ${\\mathbb X}$ can then be expanded in terms of the orthonormal system of functions $\\{\\phi_k\\}$. The extension to ${\\mathbb R}^D$ is achieved by treating \\eref{kerneleigdef} as a definition of $\\phi_k$ on ${\\mathbb R}^D$ (which can be done since $K$ is defined on ${\\mathbb R}^D\\times {\\mathbb R}^D$), and the expansion of the original function $f$ where the basis functions are now interpreted as extended by \\eref{kerneleigdef} as the desired extension. This leads to a variety of theoretical problems related to the judicious construction of kernels defined on the whole space whose eigenfunctions are meaningful as functions on ${\\mathbb X}$ (e.g., kernels that commute with the Laplace--Beltrami operator) so as to allow such a construction. In the end, it is not clear how well this extension will behave outside of ${\\mathbb X}$.\n\nOur construction gives an alternative method for extending a function on ${\\mathbb X}$ to a tubular neighborhood of ${\\mathbb X}$, which \nwe feel is more appropriate for most applications rather than trying to extend the function to the entire ambient space. Toward this goal, we first explain the local coordinate learning phase for tubular neighborhood of ${\\mathbb X}$. \n\nLet $s\\ge d$ be an integer, $s\\le D$. For $\\mathbf{q}\\in{\\mathbb R}^s$ (or $\\mathbf{q}\\in{\\mathbb R}^D$), we write $\\pi_c(\\mathbf{q})=(q^1,\\cdots,q^d)$, and\n$$\nv(\\mathbf{q})=u(\\pi_c(\\mathbf{q}))+(\\underbrace{0,\\cdots,0}_{d \\mbox{ times}},q^{d+1},\\cdots,q^s,\\underbrace{0,\\cdots,0}_{D-s \\mbox{ times}})\\in{\\mathbb R}^D.\n$$\nIf $J_s$ is the Jacobian matrix for $v$, i.e., $(J_s)_{i,j}(\\mathbf{q})=D_iv^j(\\mathbf{q})$, then it is easy to check that\n\\begin{equation}\\label{extjacobi}\nJ_s(0)=[I_s|0_{s,D-s}],\n\\end{equation}\nand\n\\begin{equation}\\label{extjacobidiff}\n(J_s(\\mathbf{q})-J_s(\\mathbf{p}))(z)=(J_d(\\pi_c(\\mathbf{q}))-J_d(\\pi_c(\\mathbf{p})))(z), \\qquad \\mathbf{p},\\mathbf{q}\\in B_d(0,\\delta^*), \\ z\\in{\\mathbb R}^D.\n\\end{equation}\nConsequently, \n\\begin{equation}\\label{extkappa}\n\\|J_s(\\mathbf{q})-J_s(0)\\|\\le \\kappa\\|\\pi_c(\\mathbf{q})\\|_d\\le \\kappa\\|\\mathbf{q}\\|_s.\n\\end{equation}\nIf we now define for $\\mathbf{p}, \\mathbf{q}\\in B_d(0,\\delta^*)$\n$$\n\\rho_1(v(\\mathbf{p}),v(\\mathbf{q}))^2:=\\rho(u(\\pi_c(\\mathbf{p})),u(\\pi_c(\\mathbf{q})))^2 +\\sum_{k=d+1}^s |p^k-q^k|^2,\n$$\nthen following the same argument as the one leading to \\eref{rhoeuccomp1} leads to\n$$\n(1\/2)\\rho_1(v(\\mathbf{p}),v(\\mathbf{q}))\\le \\|\\mathbf{p}-\\mathbf{q}\\|_s\\le 2\\rho_1(v(\\mathbf{p}), v(\\mathbf{q})), \\qquad \\|\\mathbf{p}\\|_s, \\|\\mathbf{q}\\|_s\\le \\eta^*.\n$$\n\nThus, there is no loss of generality in assuming that ${\\mathbb X}$ is already a $s$ dimensional submanifold of ${\\mathbb R}^D$, defined by $v$, and with geodesic distance $\\rho_1$. This has several consequences. Even if one overestimates the dimension of the original manifold to be $s$ rather than $d$, the resulting ``distance respecting'' coordinate system will also be ``distance respecting'' for the original manifold, except for the presumably small error resulting from the overestimate. If we have no information about $d$ (or $s$), we may take $s=D$. This would answer the question regarding points off the manifold, as well as take noise into account. However, then the advantage of dimension reduction is lost. Also, all the constants will depend upon $D$ rather than $s$ (or $d$). \n\nHaving defined a local coordinate system for the tubular neighborhood of ${\\mathbb X}$ in this way, we can then construct the local basis functions on this neighborhood as in Section~\\ref{locbasissect}.  However, since the original data ${\\mathcal C}$ is not dense on the local coordinate patch in the tubular neighborhood, one cannot use the ideas in Section~\\ref{approxsect}. We would like instead to keep some control on the growth of the extension operator. For this purpose, we propose to use the minimal Sobolev norm (MSN) interpolant introduced in \\cite{bdint} and used very fruitfully in solutions of partial differential equations \\cite{shivpdf} and image segmentation \\cite{spie10}.\n\nThus, using the procedures explained in these papers, we consider a differential operator $\\Delta$ depending upon the application, and find \nan integer $N$ and coefficients $c_{\\bf k}^*$, $|{\\bf k}|_\\infty\\le N$, so as the $s$--variate polynomial $P^*=\\sum_{{\\bf k} : |{\\bf k}|_\\infty \\le N} c_{\\bf k}^*T_{\\bf k}$ minimizes\n\\begin{equation}\\label{msndef}\n\\int_{[-1,1]^s}|\\Delta P(\\mathbf{t})|^2\\frac{d\\mathbf{t}}{\\sqrt{(1-(t^1)^2)\\cdots (1-(t^s)^2)}}\n\\end{equation}\nover all $s$--variate polynomials $P$ of coordinatewise degree $\\le N$, subject to the conditions \n\\begin{equation}\\label{intcond}\nP(\\Phi({\\bf x}_j))=f({\\bf x}_j), \\qquad {\\bf x}_j\\in{\\mathcal C}.\n\\end{equation} \nThe polynomial $P^*$ then defines an extension of $f$ to the tubular neighborhood of  the local coordinate patch of ${\\mathbb X}$.\\\\\n\n\\noindent\n\\textbf{Acknowledgments}\\\\\n\nWe would like to thank Professor Tomaso Poggio for his comments and for sharing the manuscript \\cite{poggio_deep_net_2015} with us.\n\n\n\n\\newpage\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\n\\subsection{Notation}\n\\label{sec:notations}\n\n\nOur analysis makes use of tensor algebra in both element-wise equations and specialized notation for tensor operations~\\cite{kolda2009tensor}.\nVectors are denoted with bold lowercase Roman letters (e.g., $\\vcr{v}$), matrices are denoted with bold uppercase Roman letters (e.g., $\\mat{M}$), and tensors are denoted with bold calligraphic font (e.g., $\\tsr{T}$). \nAn order $N$ tensor corresponds to an $N$-dimensional array. \nElements of vectors, matrices, and tensors are denoted in parentheses, e.g., $\\vcr{v}(i)$ for a vector $\\vcr{v}$, $\\mat{M}(i,j)$ for a matrix $\\mat{M}$, and $\\tsr{T}(i,j,k,l)$ for an order 4 tensor $\\tsr{T}$.\nThe $i$th column of $\\mat{M}$ is denoted by $\\mat{M}(:,i)$, and the $i$th row is denoted by $\\mat{M}(i,:)$. Parenthesized superscripts are used to label different vectors, matrices and tensors (e.g. $\\tsr{T}^{(1)}$ and $\\tsr{T}^{(2)}$ are unrelated tensors). Number of nonzeros of the tensor $\\tsr{T}$ is denoted by $\\nnz(\\tsr{T})$.\n\nThe pseudo-inverse of matrix $\\mat{A}$ is denoted with $\\mat{A}^\\dagger$. The Hadamard product of two matrices is denoted with $*$. The outer product of two or more vectors is denoted with $\\circ$. The Kronecker product of two vectors\/matrices is denoted with $\\otimes$.\nFor matrices $\\mat{A}\\in \\mathbb{R}^{m\\times k}$ and $\\mat{B}\\in \\mathbb{R}^{n\\times k}$, their Khatri-Rao product results in a matrix of size $(mn)\\times k$ defined by\n$\n   \\mat{A}\\odot \\mat{B} = [\\mat{A}(:,1)\\otimes \\mat{B}(:,1),\\ldots, \\mat{A}(:,k)\\otimes \\mat{B}(:,k)] .\n$\nThe mode-$n$ tensor times matrix of an order $N$ tensor $\\tsr{T} \\in \\mathbb{R}^{s_1\\times \\cdots \\times s_N}$ with a matrix $ \\textbf{A}\\in \\mathbb{R}^{J\\times s_n}$ is denoted by $\\tsr{T}\\times_n \\mat{A}$, whose output size is $s_1\\times\\cdots\\times s_{n-1}\\times J\\times s_{n+1}\\times\\cdots\\times s_N$.\nMatricization is the process of unfolding a tensor into a matrix. The mode-$n$ matricized version of $\\tsr{T}$ is denoted by $\\textbf{T}_{(n)}\\in \\mathbb{R}^{s_n\\times K}$ where $K=\\prod_{m=1,m\\neq n}^N s_m$. \nWe use $[N]$ to denote $\\{1,\\ldots, N\\}$. $\\widetilde{\\mathcal{O}}$ denotes the asymptotic cost with logarithmic factors ignored.  \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Previous Work}\n\nRandomized algorithms have been applied to both Tucker and CP decompositions in several previous works. For Tucker decomposition, Ahmadi-Asl et al. \\cite{ahmadi2020randomized} review a variety of random projection, sampling and sketching based randomized algorithms. \nMethods introduced in \\cite{che2021randomized,che2021randomized,che2019randomized,zhou2014decomposition,sun2018tensor} accelerate the traditional HOSVD\/HOOI via random projection, where factor matrices are updated based on performing SVD on the matricization of the randomly projected input tensor.\nFor these methods, random projections are all performed based on Gaussian embedding matrices, \nand the core tensor is calculated via TTMc among the input tensor and all the factor matrices, which costs $\\Omega(\\nnz(\\tsr{T})R)$ and is computationally inefficient for large sparse tensors.\nSun et al. \\cite{sun2020low} introduce randomized algorithms for Tucker decompositions for streaming data.\n\nThe most similar work to ours is Becker and Malik~\\cite{malik2018low}. \nThis work computes Tucker decomposition via a sketched ALS scheme where in each optimization subproblem, one of the factor matrices or the core tensor is updated. They also solve each sketched linear least squares subproblem via TensorSketch.\nOur new scheme provides more accurate results compared to this method. \nAnother work that is closely relevant to us is \\cite{minster2020randomized}. This work introduces structure-preserving decomposition, which is similar to Tucker decomposition but the factor matrices are not necessary orthogonal, and the entries of the core tensor are explicitly taken from the original tensor. The authors design an algorithm based on rank-revealing QR~\\cite{gu1996efficient}, which is efficient for sparse tensors, to calculate the decomposition. However, their experimental results show that the relative error of the algorithm for sparse tensors is much worse than that of the traditional HOSVD~\\cite{minster2020randomized}. \n\n\nSeveral works discuss algorithms for sparse Tucker decomposition. Oh et al.~\\cite{oh2018scalable} propose PTucker, which provides algorithms for scalable sparse Tucker decomposition. Kaya and Ucar~\\cite{kaya2016high} provide parallel algorithms for sparse Tucker decompositions. Li et al.~\\cite{li2020sgd_tucker} introduce SGD-Tucker, which uses stochastic gradient descent to perform Tucker decomposition of sparse tensors.\n\nFor CP decomposition, Battaglino et al.~\\cite{battaglino2018practical} and Jin et al.~\\cite{jin2019faster} introduce a randomized algorithm based on Kronecker fast Johnson-Lindenstrauss Transform (KFJLT) to accelerate CP-ALS. However, KFJLT is effective only for the decomposition of dense tensors.\nAggour et al.~\\cite{aggour2020adaptive} introduce adaptive sketching for CP decomposition.\nSong et al.~\\cite{song2019relative} discuss the theoretical relative error of various tensor decompositions based on sketching. \nThe work by Cheng et al.~\\cite{cheng2016spals} and Larsen and Kolda~\\cite{larsen2020practical} accelerate CP-ALS based on leverage score sampling. \nCheng et al.~\\cite{cheng2016spals} use leverage score sampling to accelerate MTTKRP calculations.\nLarsen and Kolda~\\cite{larsen2020practical} propose an approximate leverage score sampling scheme for the Khatri-Rao product, and they show with $\\bigO{R^{(N-1)}\\log(1\/\\delta)\/\\epsilon^2 }$ number of samples, each unconstrained linear least squares subproblem in CP-ALS can be solved with $\\bigO{\\epsilon}$-relative error with probability at least $1-\\delta$. \nZhou et al.~\\cite{zhou2014decomposition} and Erichson et al.~\\cite{erichson2020randomized} accelerate CP decomposition via performing randomized Tucker decomposition of the input tensor first, and then performing CP decomposition of the smaller core tensor.\n\nSeveral other works discuss techniques to parallelize and accelerate CP-ALS. Ma and Solomonik \\cite{ma2018accelerating,ma2020efficient} approximate MTTKRP within CP-ALS based on information from previous sweeps.\nFor sparse tensors, parallelization strategies for MTTKRP have been developed both on shared memory systems~\\cite{nisa2019load,smith2015splatt} and distributed memory systems~\\cite{li2017model,smith2016medium,kaya2018parallel}.\n\n\n\n\n\n\n\n\n\n\\subsection{Our Contributions}\nIn this work, we propose a new sketched\nALS algorithm for Tucker decomposition. Different from Becker and Malik~\\cite{malik2018low}, our ALS scheme is the same as HOOI, where one of the factor matrices along with the core tensor are updated in each subproblem. This guarantees the algorithm can reach the same accuracy as HOOI with sufficient sketch size. Experimental results show that it provides more accurate results compared to those in \\cite{malik2018low}.\n\nIn this scheme, each subproblem is a sketched \\textit{rank-constrained} linear least squares problem, with the left-hand-side matrix with size $s^{N-1}\\times R^{N-1}$  composed of orthonormal columns.\nTo the best of our knowledge, the relative error analysis of sketching techniques for this problem have not been discussed in the literature. Existing works either only provide sketch size upper bounds for the relaxed problem~\\cite{pilanci2016iterative}, where rank constraint is relaxed with a nuclear norm constraint, or provide upper bounds for general constrained problems, which are too loose~\\cite{woodruff2014sketching}.\nWe provide tighter sketch size upper bounds to achieve $\\bigO{\\epsilon}$ relative error with two state-of-the-art sketching techniques, TensorSketch~\\cite{pagh2013compressed} and leverage score sampling~\\cite{drineas2012fast}. \n\nWith leverage score sampling, our analysis shows that \n with probability at least $1-\\delta$,\nthe sketch size of $\\bigO{R^{N-1}\/(\\epsilon^2\\delta) }$ is sufficient for results with $\\bigO{\\epsilon}$-relative error. With TensorSketch, the sketch size upper bound is $\\bigO{(R^{N-1} \\cdot 3^{N-1})\/\\delta \\cdot (R^{N-1}  + 1\/\\epsilon^2)}$, at least $\\bigO{3^{N-1}}$ times that for leverage score sampling. For both techniques, our bounds are at most $\\bigO{1\/\\epsilon}$ times the sketch size upper bounds for the unconstrained linear least squares problem. \n\n\nThe upper bounds suggest that under the same accuracy criteria, leverage score sampling potentially needs smaller sketch size for each linear least squares problem and thus can be more efficient than TensorSketch. Therefore, with the same sketch size, the accuracy with leverage score sampling can be better. However, with the standard random initializations for factor matrices, leverage score sampling can perform poorly on tensors with high coherence~\\cite{candes2009exact} (the orthogonal basis for the row space of each matricization of the input tensor has large row norm variability), making it less robust than TensorSketch.\nTo improve the robustness of leverage score sampling, we introduce an algorithm that uses the randomized range finder (RRF)~\\cite{halko2011finding} to initialize the factor matrices. \nThe initialization scheme uses the composition of CountSketch and Gaussian random matrix as the RRF embedding matrix, which\nonly requires one pass over the non-zero elements of the input tensor. Our experimental results show that the leverage score sampling based randomized algorithm combined with this RRF scheme performs well on tensors with high coherence. \n\nFor $R\\ll s$, our new sketching based algorithm for Tucker decomposition can also be used to accelerate CP decomposition. Tucker compression is performed first, and then CP decomposition is applied to the core tensor~\\cite{zhou2014decomposition,bro1998improving,erichson2020randomized}.\nSince the per-sweep costs for both sketched Tucker-ALS and sketched CP-ALS are comparable, and Tucker-ALS often needs much fewer sweeps than CP-ALS (Tucker-ALS typically converges in less than 5 sweeps based on our experiments), this Tucker + CP method can be more efficient than directly applying randomized CP decomposition~\\cite{larsen2020practical,cheng2016spals} on the input tensor.\n\nIn summary, this paper makes the following contributions.\n\\begin{itemize}[itemsep=0pt,leftmargin=*]\n    \\item We introduce a new sketched ALS algorithm for Tucker decomposition, which contains a sequence of sketched rank-constrained linear least squares subproblems. We provide theoretical upper bounds for the sketch size, and experimental results show that the algorithm provides up to $22.0\\%$ relative decomposition residual improvement compared to the previous work.\n    \\item We provide detailed comparison of TensorSketch and leverage score sampling in terms of efficiency and accuracy. Our theoretical analysis shows that leverage score sampling is better in terms of both metrics.\n    \\item We propose an initialization scheme based on RRF that improves the accuracy of leverage score sampling based sketching algorithm on tensors with high coherence.\n    \\item We show that CP decomposition can be more efficiently and accurately calculated based on the sketched Tucker + CP method, compared to directly performing sketched CP-ALS on the input tensor.\n\\end{itemize}\n\n\n\n\n\\subsection{Leverage Score Sampling for Unconstrained \\& Rank-constrained Least Squares}\n\\label{subsec:leverage_background}\n\nIn this section, we first give the sketch size bound that is sufficient for the leverage score sampling matrix to be ab $(1\/2, \\delta, \\epsilon)$-accurate sketching matrix according to \\cref{lem:structure_leverage}. \nUsing \\cref{lem:structure_leverage}, we can then easily derive the upper bounds for both unconstrained and rank-constrained linear least squares with leverage score sampling.\nTo establish these results, we leverage two lemmas.\nLemma~\\ref{lam:matmul_leverage} bounds the sketch size sufficient to reach certain matrix multiplication accuracy, while Lemma~\\ref{lam:singular_leverage} bounds the singular values of the sketched matrix obtained from applying leverage score sampling to a matrix with orthonormal columns. \nThese first two lemmas follow from prior work, and we direct readers to references for detailed proofs of both lemmas.\n\n\n\\begin{lemma}[Approximate Matrix Multiplication with Leverage Score Sampling~\\cite{larsen2020practical}]\\label{lam:matmul_leverage}\nGiven matrices $\\mat{P}\\in\\BB{R}^{s^{N-1}\\times R^{N-1}}$ consists of orthonormal columns and $\\mat{B}\\in \\mathbb{R}^{s^{N-1}\\times n}$.\nLet $\\mat{S}\\in\\BB{R}^{m \\times s^{N-1}}$ be a leverage score sampling matrix for $\\mat{P}$.\nFor $m \\geq 1\/(\\epsilon^2\\delta) $,\nthe approximation,\n\\[\n\\|\\mat{P}^T\\mat{S}^T \\mat{SB} - \\mat{P}^T\\mat{B}\\|_F^2\\leq \\epsilon^2 \\cdot \\|\\mat{P}\\|_F^2\\cdot \\|\\mat{B}\\|_F^2,\n\\]\nholds with probability at least $1-\\delta$.\n\\end{lemma}\n\n\\begin{lemma}[Singular Value Bound for Leverage Score Sampling~\\cite{woodruff2014sketching}]\\label{lam:singular_leverage}\nGiven a full-rank matrix $\\mat{P}\\in\\BB{R}^{s^{N-1}\\times R^{N-1}}$ with $s>R$, and $\\mat{B}\\in \\mathbb{R}^{s^{N-1}\\times n}$.\nLet $\\mat{S}\\in\\BB{R}^{m \\times s^{N-1}}$ be a leverage score sampling matrix for $\\mat{P}$.\nFor $m = \\bigO{R^{(N-1)}\\log(R^{(N-1)}\/\\delta)\/\\gamma^2}= \\bigOt{R^{(N-1)}\/\\gamma^2}$,\neach singular value $\\sigma$ of $\\mat{SQ}_P$ satisfies\n\\[\n        1-\\gamma\\leq \\sigma^2 \\leq 1+\\gamma\n\\]\nwith probability at least $1-\\delta$, where  $\\mat{Q}_P$ is an orthonormal basis for the column space of $\\mat{P}$.\n\\end{lemma}\n\n\n\n\\begin{lemma}[$(1\/2, \\delta, \\epsilon)$-accurate Leverage Score Sampling Matrix]\\label{lem:structure_leverage}\nLet $m=  \\bigO{R^{N-1}\/(\\epsilon^2\\delta)}$ denote the sketch size,\nthen the leverage score sampling matrix $\\mat{S}\\in\\BB{R}^{m \\times s^{N-1}}$ is a $(1\/2, \\delta, \\epsilon)$-accurate sketching matrix for the full-rank matrix $\\mat{P}\\in\\BB{R}^{s^{N-1}\\times R^{N-1}}$.\n\\end{lemma}\n\\begin{proof}\nBased on \\cref{lam:singular_leverage} with $\\gamma=1\/2$, for $m = \\bigOt{R^{(N-1)}}$, \\eqref{eq:structure1} in \\cref{def:accurate_skeching} will hold.\nBased on \\cref{lam:matmul_leverage}, for $m= \\bigO{R^{N-1}\/(\\epsilon^2\\delta)}$, \n\\[\n\\|\\mat{Q}_P^T\\mat{S}^T \\mat{SB} - \\mat{Q}_P^T\\mat{B}\\|_F^2\\leq \\frac{\\epsilon^2}{R^{N-1}} \\cdot \\|\\mat{Q}_P\\|_F^2\\cdot \\|\\mat{B}\\|_F^2 = \\epsilon^2\\|\\mat{B}\\|_F^2,\n\\]\nthus \\eqref{eq:structure2} in \\cref{def:accurate_skeching} will hold. Thus we need\n$m= \\bigOt{R^{(N-1)}} + \\bigO{R^{N-1}\/(\\epsilon^2\\delta)}$$=\\bigO{R^{N-1}\/(\\epsilon^2\\delta)}$.\n\\end{proof}\n\n\\begin{theorem}[Leverage Score Sampling for Unconstrained Linear Least Squares]\n\\label{thm:leverage-ucls}\nGiven a full-rank matrix $\\mat{P}\\in\\BB{R}^{s^{N-1}\\times R^{N-1}}$ with $s>R$, and $\\mat{B}\\in \\mathbb{R}^{s^{N-1}\\times n}$.\nLet $\\mat{S}\\in\\BB{R}^{m \\times s^{N-1}}$ be a leverage score sampling matrix.\nLet $\\mat{\\widetilde{X}}_{\\text{opt}}= \\arg\\min_{\\mat{X}} \\fnrm{\\mat{SPX}-\\mat{SB}}$ and $\\mat{X}_{\\text{opt}} = \\arg\\min_{\\mat{X}} \\left\\|\\mat{PX}-\\mat{B}\\right\\|_F$. \nWith \n\\begin{equation}\\label{eq:size_leverage_ucls}\n  m = \\bigO{R^{N-1}\/(\\epsilon\\delta)},  \n\\end{equation}\nthe approximation,\n$\n\\left\\|\\mat{A}\\mat{\\widetilde{X}}_{\\text{opt}}- \\mat{B}\\right\\|_F^2\t \\leq \\left(1+\\bigO{\\epsilon}\\right) \\Big\\|\\mat{A}\\mat{X}_{\\text{opt}}- \\mat{B}\\Big\\|_F^2,\n$\nholds with probability at least $1-\\delta$.\n\\end{theorem}\n\\begin{proof}\nBased on \\cref{thm:structure-ls}, to prove this theorem, we derive the sample size $m$ sufficient to make the sketching matrix $(1\/2, \\delta, \\sqrt{\\epsilon})$-accurate. According to \\cref{lem:structure_leverage}, the sketch size \\eqref{eq:size_leverage_ucls} is sufficient for being $(1\/2, \\delta, \\sqrt{\\epsilon})$-accurate.\n\\end{proof}\n\n\\begin{proof}[Proof of \\cref{thm:leverage-rcls}]\nBased on \\cref{thm:structure-rcls}, to prove this theorem, we derive the sketch size $m$ sufficient to make the sketching matrix $(1\/2, \\delta, \\epsilon)$-accurate. \nAccording to \\cref{lem:structure_leverage}, the sketch size \n$\\bigO{R^{N-1}\/(\\epsilon^2\\delta)}$ is sufficient for being $(1\/2, \\delta, \\epsilon)$-accurate.\n\\end{proof}\n\n\n\n\\subsection{Error Bound for Sketched Unconstrained Linear Least Squares}\\label{subsec:general_analysis_unconstrained}\n\nWe define the $(\\gamma, \\delta, \\epsilon)$-accurate sketching matrix in \\cref{def:accurate_skeching}. \nIn \\cref{thm:structure-ls}, we show the relative error bound for the unconstrained linear least squares problem with a $(1\/2, \\delta, \\epsilon)$-accurate sketching matrix. \nBy $\\mat{Q}_P$ we denote a matrix whose columns form an orthonormal basis for the column space of $\\mat{P}$.\n\n\\begin{definition}[$(\\gamma, \\delta, \\epsilon)$-accurate Sketching Matrix]\\label{def:accurate_skeching} A random matrix $\\mat{S}\\in\\BB{R}^{m \\times s}$ is a $(\\gamma, \\delta, \\epsilon)$-accurate sketching matrix for $\\mat{P}\\in\\BB{R}^{s\\times R}$ if the following two conditions hold simultaneously.\n\\begin{enumerate}[leftmargin=*]\n    \\item With probability at least $1 - \\delta\/2$, each singular value $\\sigma$ of $\\mat{SQ}_P$ satisfies\n    \\begin{equation}\\label{eq:structure1}\n        1-\\gamma\\leq \\sigma^2 \\leq 1+\\gamma.\n    \\end{equation}\n    \\item With probability at least $1 - \\delta\/2$, for any fixed matrix $\\mat{B}$, we have\n    \\begin{equation}\\label{eq:structure2}\n        \\|\\mat{Q}_P^T\\mat{S}^T \\mat{SB} - \\mat{Q}_P^T\\mat{B}\\|_F^2\\leq \\epsilon^2 \\cdot \\|\\mat{B}\\|_F^2.\n    \\end{equation}\n\\end{enumerate} \n\\end{definition}\n\n\\begin{lemma}[Linear Least Squares with $(1\/2, \\delta, \\epsilon)$-accurate Sketching Matrix~\\cite{woodruff2014sketching,larsen2020practical,drineas2011faster}]\n\\label{thm:structure-ls}\nGiven a full-rank matrix $\\mat{P}\\in\\BB{R}^{s\\times R}$ with $s\\geq R$, and $\\mat{B}\\in \\mathbb{R}^{s\\times n}$. \nLet $\\mat{S}\\in\\BB{R}^{m\\times s}$ be a $(1\/2, \\delta, \\epsilon)$-accurate sketching matrix.\nLet $\\mat{B}^{\\perp} = \\mat{P}\\mat{X}_{\\text{opt}} - \\mat{B}$, with $\\mat{X}_{\\text{opt}} = \\arg\\min_{\\mat{X}} \\left\\|\\mat{PX}-\\mat{B}\\right\\|_F$, and $\\mat{\\widetilde{X}}_{\\text{opt}} = \\arg\\min_{\\mat{X}} \\fnrm{\\mat{SPX}-\\mat{SB}}$.\nThen the following approximation holds with probability at least $1-\\delta$,\n\t\\begin{equation}\n\\left\\|\\mat{P}\\mat{\\widetilde{X}}_{\\text{opt}}- \\mat{P}\\mat{X}_{\\text{opt}}\\right\\|_F^2\t \n\\leq \\bigO{\\epsilon^2} \\left\\|\\mat{B}^{\\perp}\\right\\|_F^2.\n\t\\end{equation}\t\n\\end{lemma}\n\\begin{proof}\nDefine the reduced QR decomposition,\n$\\mat{P} = \\mat{Q}_{P}\\mat{R}_P$. \nThe unconstrained sketched problem can be rewritten as\n\\begin{align*}\n   \\min_{\\mat{X}} \\left\\|\\mat{SPX}-\\mat{SB}\\right\\|_F \n   &= \n\\min_{\\mat{X}} \\left\\|\\mat{SPX}-\\mat{S}(\\mat{P}\\mat{X}_{\\text{opt}} + \\mat{B}^{\\perp})\\right\\|_F \\\\\n&= \n\\min_{\\mat{X}} \\left\\|\\mat{S}\\mat{Q}_P\\mat{R}_P(\\mat{X}-\\mat{X}_{\\text{opt}}) -\\mat{S}\\mat{B}^{\\perp}\\right\\|_F, \n\\end{align*}\nthus the optimality condition is\n\\begin{equation}\\label{eq:opt_condition}\n(\\mat{S}\\mat{Q}_P)^T\\mat{S}\\mat{Q}_P\\mat{R}_P(\\mat{\\widetilde{X}}_{\\text{opt}}-\\mat{X}_{\\text{opt}}) = (\\mat{S}\\mat{Q}_P)^T\\mat{S}\\mat{B}^{\\perp}.   \n\\end{equation}\nBased on \\eqref{eq:structure1},\\eqref{eq:structure2}, with probability at least $1 - \\delta$,\nboth of the following hold,\n\\begin{gather}\\label{eq:structure1_theorema1}\n    \\sigma_{\\min}^2 (\\mat{S}\\mat{Q}_P) \\geq 1 - \\gamma = 1\/2,\\\\\n\\label{eq:structure2_theorema1}\n    \\left\\| \\mat{Q}_P^T \\mat{S}^T \\mat{S} \\mat{B}^{\\perp} \\right\\|_F^2 \n    = \n    \\left\\|\\mat{Q}_P^T\\mat{S}^T \\mat{S}\\mat{B}^{\\perp} - \\mat{Q}_P^T\\mat{B}^{\\perp}\\right\\|_F^2 \\leq \\epsilon^2 \\cdot \\left\\|\\mat{B}^{\\perp}\\right\\|_F^2,\n\\end{gather}\nwhere $\\sigma_{\\min} (\\mat{S}\\mat{Q}_P)$ is the singular value of $\\mat{S}\\mat{Q}_P$ with the smallest magnitude.\nCombining \\eqref{eq:opt_condition}, \\eqref{eq:structure1_theorema1}, and \\eqref{eq:structure2_theorema1}, we obtain\n\\begin{align*}\n\\left\\|\\mat{P}\\mat{\\widetilde{X}}_{\\text{opt}}\n-\\mat{P}\\mat{X}_{\\text{opt}}\\right\\|_F^2\n&=\n\\left\\|\\mat{R}_P\\mat{\\widetilde{X}}_{\\text{opt}}\n-\\mat{R}_P\\mat{X}_{\\text{opt}}\\right\\|_F^2 \\\\\n&\\overset{\\eqref{eq:structure1_theorema1}}{\\leq}\n4\\left\\|\n(\\mat{S}\\mat{Q}_P)^T\\mat{S}\\mat{Q}_P\\mat{R}_P(\\mat{\\widetilde{X}}_{\\text{opt}}-\\mat{X}_{\\text{opt}})\n\\right\\|_F^2 \\\\\n&\\overset{\\eqref{eq:opt_condition}}{=}\n4\\left\\|\\mat{Q}_P^T \\mat{S}^T \\mat{S}\\mat{B}^{\\perp}\\right\\|_F^2 \\\\\n&\\overset{\\eqref{eq:structure2_theorema1}}{\\leq} \n4\\epsilon^2  \\cdot \\left\\|\\mat{B}^{\\perp}\\right\\|_F^2 = \\bigO{\\epsilon^2} \\left\\|\\mat{B}^{\\perp}\\right\\|_F^2.\n\\end{align*}\n\\end{proof}\n\n\\subsection{Error Bound for Sketched Rank-constrained Linear Least Squares}\\label{subsec:general_analysis}\nWe show in \\cref{thm:structure-rcls} that with at least $1-\\delta$ probability, the relative residual norm error for the rank-constrained linear least squares with a $(1\/2, \\delta, \\epsilon)$-accurate sketching matrix is  bounded by $\\bigO{\\epsilon}$.\nWe first state Mirsky's Inequality below, which bounds the perturbation of singular values when the input matrix is perturbed. We direct readers to the reference for its proof. This bound will be used in \\cref{thm:structure-rcls}.\n\\begin{lemma}[Mirsky's Inequality for Perturbation of Singular Values~\\cite{mirsky1960symmetric}]\\label{thm:perturb-value-multiple}\nLet $\\mat{A}$ and $\\mat{F}$  be arbitrary matrices (of the same size) where $\\sigma_1 \\geq \\cdots \\geq \\sigma_n$ are the singular values of $\\mat{A}$ and $\\sigma_1' \\geq \\cdots \\geq \\sigma_n'$ are the singular values of $\\mat{A} + \\mat{F}$. Then \n\\begin{equation}\\label{eq:mirsky}\n  \\sum_{i=1}^n(\\sigma_i - \\sigma_i')^2 \\leq \\|\\mat{F}\\|_F^2.  \n\\end{equation}\n\\end{lemma}\n\n\\begin{theorem}[Rank-constrained Linear Least Squares with $(1\/2, \\delta, \\epsilon)$-accurate Sketching Matrix]\n\\label{thm:structure-rcls}\nGiven $\\mat{P}\\in\\BB{R}^{s\\times R}$ with orthonormal columns (such that $\\mat{P} = \\mat{Q}_P$), and $\\mat{B}\\in \\mathbb{R}^{s\\times n}$.\nLet $\\mat{S}\\in\\BB{R}^{m\\times s}$ be a $(1\/2, \\delta, \\epsilon)$-accurate sketching matrix.\nLet $\\mat{\\widetilde{X}}_{\\text{r}}$ be the best rank-$r$ approximation of the solution of the problem\n$\\min_{\\mat{X}} \\fnrm{\\mat{SPX}-\\mat{SB}}$, and let $\\mat{X}_{\\text{r}} = \\arg\\min_{\\mat{X},\\text{rank}(\\mat{X})=r} \\left\\|\\mat{PX}-\\mat{B}\\right\\|_F$.\nThen the residual norm error bound,\n\t\\begin{equation}\n\\left\\|\\mat{P}\\mat{\\widetilde{X}}_{\\text{r}}- \\mat{B}\\right\\|_F^2\t \n\\leq \\left(1+\\bigO{\\epsilon}\\right) \\Big\\|\\mat{P}\\mat{X}_{\\text{r}}- \\mat{B}\\Big\\|_F^2\t,\n\t\\end{equation}\t\nholds with probability at least $1-\\delta$.\n\\end{theorem}\n\\begin{proof}\nLet $\\mathcal{R} = \\left\\|\\mat{P}\\mat{X}_{\\text{r}}\n- \\mat{B}\\right\\|_F$. In addition, let  $\\mat{X}_{\\text{opt}} = \\arg\\min_{\\mat{X}} \\left\\|\\mat{PX}-\\mat{B}\\right\\|_F$ be the optimum solution of the unconstrained linear least squares problem.\nSince the residual in the true solution for each component of the least-squares problem (column of $\\mat{B}^{\\perp}$) is orthogonal to the error due to low rank approximation,\n\\begin{align}\n    \\mathcal{R}^2 = \\left\\|\\mat{P}\\mat{X}_{\\text{r}}\n- \\mat{B}\\right\\|^2_F &= \\left\\|\\mat{P}\\mat{X}_{\\text{opt}}\n- \\mat{B}\\right\\|^2_F + \\left\\|\\mat{P}\\mat{X}_{\\text{r}}\n- \\mat{P}\\mat{X}_{\\text{opt}}\\right\\|^2_F \\nonumber\\\\\n& = \\left\\|\\mat{B}^{\\perp}\\right\\|_F^2 + \\left\\|\\mat{X}_{\\text{r}}\n- \\mat{X}_{\\text{opt}}\\right\\|^2_F.\n\\end{align}\nThe last equality holds since $\\mat{P}$ has orthonormal columns. \nLet $\\mat{\\widetilde{X}}_{\\text{opt}} = \\arg\\min_{\\mat{X}} \\left\\|\\mat{SPX}-\\mat{SB}\\right\\|_F$ be the optimum solution of the unconstrained sketched problem. We have\n\\begin{align}\n    \\left\\|\\mat{P}\\mat{\\widetilde{X}}_{\\text{r}}- \\mat{B}\\right\\|_F^2\t\n    &= \\left\\|\\mat{P}\\mat{\\widetilde{X}}_{\\text{r}}- \\mat{P}\\mat{\\widetilde{X}}_{\\text{opt}}\\right\\|_F^2\t+ \\left\\|\\mat{P}\\mat{\\widetilde{X}}_{\\text{opt}}- \\mat{B}\\right\\|_F^2\n    + 2 \\left\\langle \\mat{P}\\mat{\\widetilde{X}}_{\\text{r}}- \\mat{P}\\mat{\\widetilde{X}}_{\\text{opt}},\n    \\mat{P}\\mat{\\widetilde{X}}_{\\text{opt}}-\\mat{B}\n    \\right\\rangle_F \\notag\\\\\n    \\label{eq:decompose_error}\n&= \\left\\|\\mat{\\widetilde{X}}_{\\text{r}}-\\mat{\\widetilde{X}}_{\\text{opt}}\\right\\|_F^2\t+ \n\\left\\|\\mat{\\widetilde{X}}_{\\text{opt}}\n-\\mat{X}_{\\text{opt}}\\right\\|_F^2\n+ \\left\\|\\mat{B}^{\\perp}\\right\\|_F^2\n    + 2 \\left\\langle \\mat{\\widetilde{X}}_{\\text{r}}- \\mat{\\widetilde{X}}_{\\text{opt}},\n    \\mat{\\widetilde{X}}_{\\text{opt}}-\\mat{X}_{\\text{opt}}\n    \\right\\rangle_F.\n\\end{align}\nNext we bound the magnitudes of the first, second and the fourth terms.\nAccording to \\cref{thm:structure-ls}, with probability at least $1-\\delta$, the second term in \\eqref{eq:decompose_error} can be bounded as\n\\begin{equation}\\label{eq:fullrank_error}\n  \\left\\|\\mat{\\widetilde{X}}_{\\text{opt}}\n-\\mat{X}_{\\text{opt}}\\right\\|_F^2\n=\n\\left\\|\\mat{P}\\mat{\\widetilde{X}}_{\\text{opt}}\n-\\mat{P}\\mat{X}_{\\text{opt}}\\right\\|_F^2\n\\leq C\\epsilon^2 \\left\\|\\mat{B}^{\\perp}\\right\\|_F^2,\n\\end{equation}\nfor some constant $C\\geq 1$.\nSuppose $ \\mat{\\widetilde{X}}_{\\text{opt}}$ has singular values $\\widetilde{\\sigma}_i = \\sigma_i + \\delta \\sigma_i$ for $i$ in $\\{1, \\ldots, \\min(R,n)\\}$, where $\\sigma_i$ are the singular values of\n$\\mat{X}_{\\text{opt}}$.\nSince $\\mat{\\widetilde{X}}_{\\text{r}}$ is defined to be the best low rank approximation to $\\mat{\\widetilde{X}}_\\text{opt}$, we have\n\\begin{equation}\\label{eq:lowrank_error}\n    \\left\\|\\mat{\\widetilde{X}}_{\\text{r}}- \\mat{\\widetilde{X}}_{\\text{opt}}\\right\\|_F^2\n    = \\sum_{i=r+1}^{\\min(R,n)}\\widetilde{\\sigma}_i^2 =  \n    \\sum_{i=r+1}^{\\min(R,n)}(\\sigma_i + \\delta \\sigma_i)^2\n    = \\sum_{i=r+1}^{\\min(R,n)}\\left(\\sigma_i^2 + \\delta \\sigma_i^2\n    + 2\\sigma_i\\delta \\sigma_i\n    \\right).\n\\end{equation}\nSince $\\mat{P}$ has orthonormal columns, $\\mat{X}_{\\text{r}}$ is the best rank-$r$ approximation of $\\mat{X}_{\\text{opt}}$,\n\\[\n\\sum_{i=r+1}^{\\min(R,n)}\\sigma_i^2 = \\left\\|\\mat{X}_{\\text{r}}\n- \\mat{X}_{\\text{opt}}\\right\\|^2_F .\n\\]\nIn addition, based on Mirsky's inequality (\\cref{thm:perturb-value-multiple}), \n\\begin{equation}\\label{eq:delta_sigma}\n  \\sum_{i=r+1}^{\\min(R,n)}\\delta \\sigma_i^2 \n\\leq \\sum_{i=1}^{\\min(R,n)}\\delta \\sigma_i^2 \\overset{\\eqref{eq:mirsky}}{\\leq}  \\left\\|\\mat{\\widetilde{X}}_{\\text{opt}}\n-\\mat{X}_{\\text{opt}}\\right\\|_F^2\n\\overset{\\eqref{eq:fullrank_error}}{\\leq}  C\\epsilon^2 \\left\\|\\mat{B}^{\\perp}\\right\\|_F^2,  \n\\end{equation}\nand\n\\begin{align*}\n  \\sum_{i=r+1}^{\\min(R,n)}  \\left|2\\sigma_i\\delta \\sigma_i\\right|\n&= \\epsilon\\sum_{i=r+1}^{\\min(R,n)}  \\left|2\\sigma_i \\frac{\\delta \\sigma_i}{\\epsilon}\\right| \n \\leq \n\\epsilon \\sum_{i=r+1}^{\\min(R,n)} \\left(\n\\sigma_i^2 + \\frac{\\delta \\sigma_i^2}{\\epsilon^2}\n\\right) \\\\\n &\\overset{\\eqref{eq:delta_sigma}}{\\leq}  C\\epsilon\\left(\\left\\|\\mat{X}_{\\text{r}}\n- \\mat{X}_{\\text{opt}}\\right\\|^2_F + \\left\\|\\mat{B}^{\\perp}\\right\\|_F^2\n\\right) = C\\epsilon \\mathcal{R}^2,  \n\\end{align*}\nthus\n\\eqref{eq:lowrank_error} can be bounded as\n\\begin{align}\\label{eq:lowrank_error_bound}\n    \\left\\|\\mat{\\widetilde{X}}_{\\text{r}}- \\mat{\\widetilde{X}}_{\\text{opt}}\\right\\|_F^2 &\\leq \n\\left\\|\\mat{X}_{\\text{r}}\n- \\mat{X}_{\\text{opt}}\\right\\|^2_F + C\\epsilon^2 \\left\\|\\mat{B}^{\\perp}\\right\\|_F^2 + C\\epsilon \\mathcal{R}^2 \\nonumber\\\\\n&= \\left\\|\\mat{X}_{\\text{r}}\n- \\mat{X}_{\\text{opt}}\\right\\|^2_F + \\bigO{\\epsilon}\\mathcal{R}^2.\n\\end{align}\nNext we bound the magnitude of the inner product term in \\eqref{eq:decompose_error},\n\\begin{align}\n        \\left|\\left\\langle \\mat{\\widetilde{X}}_{\\text{r}}- \\mat{\\widetilde{X}}_{\\text{opt}},\n    \\mat{\\widetilde{X}}_{\\text{opt}}-\\mat{X}_{\\text{opt}}\n    \\right\\rangle_F\\right|\n    & \\leq \n    \\left\\|\\mat{\\widetilde{X}}_{\\text{r}}-\\mat{\\widetilde{X}}_{\\text{opt}}\\right\\|_F\t\\left\\|\\mat{\\widetilde{X}}_{\\text{opt}}\n    -\\mat{X}_{\\text{opt}}\\right\\|_F \\nonumber\\\\\n    & \\overset{\\eqref{eq:fullrank_error}}{\\leq}  \\sqrt{C}\\epsilon \\left\\|\\mat{\\widetilde{X}}_{\\text{r}}-\\mat{\\widetilde{X}}_{\\text{opt}}\\right\\|_F\\|\\mat{B}^{\\perp}\\|_F \\nonumber\\\\\n    &    \\leq \n    \\sqrt{C}\\frac{\\epsilon}{2}\n    \\left(\\left\\|\\mat{\\widetilde{X}}_{\\text{r}}-\\mat{\\widetilde{X}}_{\\text{opt}}\\right\\|_F^2 + \n    \\left\\|\\mat{B}^{\\perp}\\right\\|_F^2\n    \\right) \\nonumber\\\\\n    & \\overset{\\eqref{eq:lowrank_error_bound}}{\\leq}     \\sqrt{C}\\frac{\\epsilon}{2}\n    \\left(\n \\left\\|\\mat{X}_{\\text{r}}\n- \\mat{X}_{\\text{opt}}\\right\\|^2_F + \\bigO{\\epsilon}\\mathcal{R}^2\n    + \n    \\left\\|\\mat{B}^{\\perp}\\right\\|_F^2\n    \\right) \\nonumber \\\\\n    &     = \\bigO{\\epsilon}\\mathcal{R}^2.\n    \\label{eq:cross_term_bound}\n\\end{align}\nTherefore, based on \\eqref{eq:decompose_error},\\eqref{eq:fullrank_error},\\eqref{eq:lowrank_error_bound},\\eqref{eq:cross_term_bound}, with probability at least $1-\\delta$,\n\\[\n\\left\\|\\mat{P}\\mat{\\widetilde{X}}_{\\text{r}}- \\mat{B}\\right\\|_F^2\t \n\\leq \\left(1+\\bigO{\\epsilon}\\right)\\mathcal{R}^2\n= \\left(1+\\bigO{\\epsilon}\\right) \\Big\\|\\mat{P}\\mat{X}_{\\text{r}}- \\mat{B}\\Big\\|_F^2.\n\\]\n\\end{proof}\n\n\n\n\n\\subsection{TensorSketch}\n\nTensorSketch is a special type of CountSketch, where the hash map is restricted to a specific format to allow fast multiplication of the sketching matrix with the chain of Kronecker products. We introduce the definition of CountSketch and TensorSketch below.\n\n\\begin{definition}[CountSketch]\\label{def:countsketch}\nThe CountSketch matrix is defined as $\\mat{S} = \\mat{\\Omega D}\\in \\BB{R}^{m\\times n}$, where\n\\begin{itemize}[itemsep=0pt,leftmargin=*]\n    \\item $h : [n] \\rightarrow [m]$ is a hash map such that $\\forall i \\in [n]$ and $\\forall j \\in [m]$, $\\Pr[h(i) = j] = 1\/m$,\n    \\item $\\mat{\\Omega} \\in \\BB{R}^{m\\times n}$ is a matrix with $\\mat{\\Omega}(j,i) = 1$ if $j=h(i)$ $\\forall i\\in[n]$ and $\\mat{\\Omega}(j,i) = 0$ otherwise,\n    \\item $\\mat{D} \\in \\BB{R}^{n\\times n}$ is a diagonal matrix whose diagonal is a Rademacher vector (each entry is $+1$ or $-1$ with equal probability).\n\\end{itemize}\n\\end{definition}\n\n\\begin{definition}[TensorSketch~\\cite{pagh2013compressed}]\nThe order $N$ TensorSketch matrix $\\mat{S}= \\mat{\\Omega D}\\in \\BB{R}^{m \\times \\prod_{i=1}^N s_i}$ is defined based on two hash maps $H$ and $S$ defined below,\n\\begin{equation}\\label{eq:h_ts}\n    H : [s_1] \\times [s_2] \\times \\cdots \\times [s_N] \\rightarrow [m] : (i_1, \\ldots , i_N ) \\mapsto\n\\left(\n\\sum_{n=1}^N\n(H_n(i_n) - 1) \\mod m\n\\right) + 1,\n\\end{equation}\n\\vspace{-3mm}\n\\begin{equation}\\label{eq:s_ts}\n    S : [s_1] \\times [s_2] \\times \\cdots \\times [s_N] \\rightarrow \\{-1, 1\\} : (i_1, \\ldots , i_N ) \\mapsto\n    \\prod_{n=1}^N S_n(i_n),\n\\end{equation}\nwhere each $H_n$ for $n\\in[N]$ is a 3-wise independent hash map that maps $[s_n]\\rightarrow[m]$, and each $S_n$ is a 4-wise independent hash map that maps $[s_n]\\rightarrow\\{-1,1\\}$.\nA hash map is $k$-wise independent if any designated $k$ keys are independent random variables.\nTwo matrices $\\mat{\\Omega}$ and $\\mat{D}$ are defined based on $H$ and $S$, respectively,\n\\begin{itemize}[itemsep=0pt,leftmargin=*]\n    \\item $\\mat{\\Omega} \\in \\BB{R}^{m\\times \\prod_{i=1}^N s_i}$ is a matrix with $\\mat{\\Omega}(j,i) = 1$ if $j=H(i)$ $\\forall i\\in\\left[\\prod_{i=1}^N s_i\\right]$, and $\\mat{\\Omega}(j,i) = 0$ otherwise,\n    \\item $\\mat{D} \\in \\BB{R}^{n\\times n}$ is a diagonal matrix with $\\mat{D}(i,i) = S(i)$.\n\\end{itemize}\nAbove we use the notation $S(i) = S(i_1,\\dots,i_N)$ where $i = i_1 + \\sum_{k=2}^N \\left(\\prod_{\\ell=1}^{k-1} s_l\\right) \\left(i_k-1\\right)$, and similar for $H$.\n\\end{definition}\n\nThe restricted hash maps \\eqref{eq:h_ts},\\eqref{eq:s_ts} used in $\\mat{S}$ make it efficient to multiply with a chain of Kronecker products. \nDefine $\\mat{S}^{(n)}:= \\mat{\\Omega}^{(n)}\\mat{D}^{(n)} \\in \\BB{R}^{m \\times s_n}$, where $\\mat{\\Omega}^{(n)} \\in \\BB{R}^{m \\times s_n}$ is defined based on $H_n$ and $\\mat{D}^{(n)} \\in \\BB{R}^{s_n \\times s_n}$ defined based on $S_n$, and\nlet $\\mat{P} = \\mat{A}^{(1)}\\otimes \\mat{A}^{(2)}\\otimes \\cdots \\otimes \\mat{A}^{(N)}$ with $\\mat{A}^{(n)}\\in\\BB{R}^{s_n \\times R_n}$ for $n\\in[N]$, \n\\begin{equation}\n    \\mat{SP}  = \\text{FFT}^{-1}\\left(\n    \\left(\n    \\bigodot_{n=1}^N \\left(\n    \\text{FFT}\\left(\\mat{S}^{(n)}\\mat{A}^{(n)}\\right)\n    \\right)^T\n    \\right)^T\n    \\right).\n\\end{equation}\nCalculating each $\\text{FFT}\\left(\\mat{S}^{(n)}\\mat{A}^{(n)}\\right)$ costs $\\bigO{s_nR_n + m\\log m R_n}$, and performing the Kronecker product as well as the outer FFT costs $\\bigO{m\\log m\\prod_{n=1}^NR_n}$. When each $s_n=s$ and $R_n=R$, the overall cost is $\\bigO{NsR + m\\log mR^N}$.\n\n\\subsection{Leverage Score Sampling}\n\nLeverage score sampling is a useful tool to pick important rows to form the sampled\/sketched linear least squares problem. \nIntuitively, let $\\mat{Q}_P$ be an orthogonal basis for the column space of $\\mat{P}$. Then large-norm rows of $\\mat{Q}_P$ suggest large contribution to $\\mat{Q}_P^T\\mat{B}$, which is part of the linear least squares right-hand-side we can solve for.\n\n\\begin{definition}[Leverage Scores \\cite{drineas2012fast,mahoney2011randomized}]\\label{def:leverages_scores}\n  Let $\\mat{P} \\in \\BB{R}^{s \\times R}$ with $s > R$, and\n  let $\\mat{Q} \\in \\BB{R}^{s \\times R}$ be any orthogonal basis for the column space of $\\mat{P}$.\n  The \\emph{leverage scores} of the rows of $\\mat{P}$ are given by\n  \\begin{displaymath}\n    \\ell_i(\\mat{P}) := (\\mat{QQ}^T)(i,i) = \\|\\mat{Q}(i,:)\\|_2^2 \\qtext{for all} i \\in [s].\n  \\end{displaymath}\n\\end{definition}\n\n\\begin{definition}[Importance Sampling based on Leverage Scores]\\label{def:leverage_sampling} Let $\\mat{P}\\in\\BB{R}^{s\\times R}$ be a full-rank matrix and $s>R$.\nThe leverage score sampling matrix of $\\mat{P}$ is defined as $\\mat{S} = \\mat{D\\Omega}$, where $\\mat{\\Omega}\\in\\BB{R}^{m\\times s}$, $m<s$ is the sampling matrix, and $\\mat{D}\\in\\BB{R}^{m \\times m}$ is the rescaling matrix.\nFor each row $j\\in[m]$ of $\\mat{\\Omega}$,  one column index $i \\in [s]$ is picked independently with replacement with probability $p_i = \\ell_i(\\mat{P}) \/ R$, and we set $\\mat{\\Omega}(j,i) = 1, \\mat{D}(j,j) = \\frac{1}{\\sqrt{m p_i}}$. Other elements of $\\mat{\\Omega}, \\mat{D}$ are 0.\n\\end{definition}\n\nTo calculate the leverage scores of $\\mat{P}$, one can get the matrix $\\mat{Q}$ via QR decomposition, and the scores can be retrieved via calculating the norm of each row of $\\mat{Q}$.\nHowever, performing QR decomposition of $\\mat{P}$ is almost as costly as solving the linear least squares problem. The lemma below shows that leverage scores of $\\mat{P}$ can be efficiently calculated\nfrom smaller QR decompositions of the Kronecker product factors composing $\\mat{P}$.\n\n\n\\begin{lemma}[Leverage Scores for Kronecker product~\\cite{cheng2016spals}] \\label{lem:lev_bound}\n  Let $\\mat{P} = \\mat{A}^{(1)} \\otimes \\cdots \\otimes \\mat{A}^{(N)} \\in \\BB{R}^{s^N \\times R^N}$, where $\\mat{A}^{(i)}\\in \\BB{R}^{s \\times R}$ and $s>R$.\n    Leverage scores of $\\mat{P}$ satisfy\n    \\begin{equation}\\label{score_bound}\n        \\ell_i(\\mat{P}) = \\prod_{k=1}^N \\ell_{i_k}(\\mat{A}^{(k)}), \\quad\n        \\text{where } i = 1+\\sum_{k=1}^N (i_k-1) s^{k-1}.\n    \\end{equation}\n\\end{lemma}\n\\begin{proof}\nTo show \\eqref{score_bound}, we only need to show the case when $N=2$, since it can then be easily generalized to arbitrary $N$.\nConsider the reduced QR decomposition of $\\mat{A}^{(1)} \\otimes \\mat{A}^{(2)}$,\n\\begin{equation*}\n\\mat{A}^{(1)} \\otimes \\mat{A}^{(2)} = \\mat{Q}^{(1)}\\mat{R}^{(1)} \\otimes \\mat{Q}^{(2)}\\mat{R}^{(2)} = (\\mat{Q}^{(1)} \\otimes \\mat{Q}^{(2)})(\\mat{R}^{(1)} \\otimes \\mat{R}^{(2)}) =  \\mat{Q}\\mat{R}.\n\\end{equation*}\nThe reduced $\\mat{Q}$ term for $\\mat{A}^{(1)} \\otimes \\mat{A}^{(2)}$ is $ \\mat{Q}^{(1)} \\otimes \\mat{Q}^{(2)}$. Therefore, the leverage score of the $i$th row in $\\mat{Q}$, $\\ell_i$, can be expressed as,\n\\begin{align*}\n  \\ell_i(\\mat{P}) &= \\left\\|\\mat{Q}(i,:)\\right\\|_2^2\n= \\left\\|\\mat{Q}^{(1)}(i_1,:) \\otimes \\mat{Q}^{(2)}(i_2,:)\\right\\|_2^2 \\\\\n&= \\left\\|\\mat{Q}^{(1)}(i_1,:)\\right\\|_2^2 \\left\\|\\mat{Q}^{(2)}(i_2,:)\\right\\|_2^2 = \\ell_{i_1}(\\mat{A}^{(1)})\\ell_{i_2}(\\mat{A}^{(2)}).   \n\\end{align*} \n\\end{proof}\nLet $p_i = \\ell_i(\\mat{P})\/R^N$ denote the leverage score sampling probability for $i$th index, and $p^{(k)}_{i_k} = \\ell_{i_k}(\\mat{A}^{(k)})\/R$ for $k\\in[N]$ denote the leverage score sampling probability for $i_k$th index of $\\mat{A}^{(k)}$. Based on \\cref{lem:lev_bound}, we have\n\\[\np_i = p^{(1)}_{i_1} \\cdots p^{(N)}_{i_N}.\n\\]\nTherefore, leverage score sampling can be efficiently performed\nby sampling the  row of $\\mat P$ associated with multi-index $(i_1,\\dots,i_N)$, where $i_k$ is selected with probability $p^{(k)}_{i_k}$.\nTo calculate the leverage scores of each $\\mat{A}^{(k)}$,\n$N$ QR decompositions are needed, which in total cost\n $\\bigO{NsR^2}$. \nIn addition, the cost of this sampling process would be $\\bigO{Nm}$ if $m$ samples are needed, making the overall cost \n$\\bigO{NsR^2 + Nm}$.\nTo calculate $\\mat{SP}$, for each sampled multi-index $(i_1,\\dots,i_N)$, we need to perform the Kronecker product,\n\\[\n\\mat{A}^{(1)}(i_1,:) \\otimes \\cdots \\otimes \\mat{A}^{(N)}(i_N,:),\n\\]\nwhich costs $\\bigO{R^N}$.\nTherefore, including the cost of QR decompositions, overall cost is\n$\\bigO{NsR^2 + mR^N}$.\n\nRather than performing importance random sampling based on leverage scores, another way introduced in \\cite{jolliffe1972discarding} to construct the sketching matrix is to deterministically sample rows having the largest leverage scores. This idea is also used in \\cite{larsen2020practical} for randomized CP decomposition. Papailiopoulos et al.~\\cite{papailiopoulos2014provable} show that if the leverage scores follow a moderately steep power-law decay, then deterministic sampling can be provably as efficient and even better than random sampling. We compare both leverage score sampling techniques in \\cref{sec:exp}. For the sampling complexity analysis in \\cref{sec:sketch-rcls} and \\cref{sec:cost}, we only consider the random sampling technique. \n\n\\subsection{TensorSketch for Unconstrained \\& Rank-constrained Least Squares}\n\\label{subsec:tensorsketch_background}\n\nIn this section, we first give the sketch size bound that is sufficient for the TensorSketch matrix to be the $(1\/2, \\delta, \\epsilon)$-accurate sketching matrix in \\cref{lem:structure_tensorsketch}. The proof is based on \\cref{lam:matmul_tensorsketch} and \\cref{lam:singular_tensorsketch}, which follows from results derived in previous work~\\cite{avron2016sharper,diao2018sketching}.\nLemma~\\ref{lam:matmul_tensorsketch} bounds the sketch size sufficient to reach certain matrix multiplication accuracy, while Lemma~\\ref{lam:singular_tensorsketch} bounds the singular values of the matrix obtained from applying TensorSketch to a matrix with orthonormal columns. \nWe direct readers to prior work for a detailed proof of \\cref{lam:matmul_tensorsketch}, but provide a simple proof of \\cref{lam:singular_tensorsketch} by application of \\cref{lam:matmul_tensorsketch}.\n\n\n\\begin{lemma}[Approximate Matrix Multiplication with TensorSketch~\\cite{avron2016sharper}]\\label{lam:matmul_tensorsketch}\nGiven matrices $\\mat{P}\\in\\BB{R}^{s^{N-1}\\times R^{N-1}}$ and $\\mat{B}\\in \\mathbb{R}^{s^{N-1}\\times n}$.\nLet $\\mat{S}\\in\\BB{R}^{m \\times s^{N-1}}$ be an order $N-1$ TensorSketch matrix.\nFor $m \\geq (2 + 3^{N-1})\/(\\epsilon^2\\delta) $,\nthe approximation,\n\\[\n\\|\\mat{P}^T\\mat{S}^T \\mat{SB} - \\mat{P}^T\\mat{B}\\|_F^2\\leq \\epsilon^2 \\cdot \\|\\mat{P}\\|_F^2\\cdot \\|\\mat{B}\\|_F^2,\n\\]\nholds with probability at least $1-\\delta$.\n\\end{lemma}\n\n\\begin{lemma}[Singular Value Bound for TensorSketch~\\cite{diao2018sketching}]\\label{lam:singular_tensorsketch}\nGiven a full-rank matrix $\\mat{P}\\in\\BB{R}^{s^{N-1}\\times R^{N-1}}$ with $s>R$, and $\\mat{B}\\in \\mathbb{R}^{s^{N-1}\\times n}$.\nLet $\\mat{S}\\in\\BB{R}^{m \\times s^{N-1}}$ be an order $N-1$ TensorSketch matrix.\nFor $m \\geq R^{2(N-1)}(2 + 3^{N-1})\/(\\gamma^2\\delta) $,\neach singular value $\\sigma$ of $\\mat{SQ}_P$ satisfies\n\\[\n        1-\\gamma\\leq \\sigma^2 \\leq 1+\\gamma\n\\]\nwith probability at least $1-\\delta$, where  $\\mat{Q}_P$ is an orthonormal basis for the column space of $\\mat{P}$.\n\\end{lemma}\n\\begin{proof}\nSince $\\mat{Q}_P$ is an orthonormal basis for $\\mat{P}$, $\\mat{Q}^T_P\\mat{Q}_P=\\mat{I}$, and $\\|\\mat{Q}_P\\|_F^2 = R^{N-1}$. Based on \\cref{lam:matmul_tensorsketch}, for $m \\geq R^{2(N-1)}(2 + 3^{N-1})\/(\\gamma^2\\delta)$, with probability at least $1 - \\delta$, we have\n\\[\n\\left\\|\\mat{Q}_P^T\\mat{S}^T \\mat{S}\\mat{Q}_P - \\mat{Q}_P^T\\mat{Q}_P\\right\\|_F^2\n= \n\\left\\|\\mat{Q}_P^T\\mat{S}^T \\mat{S}\\mat{Q}_P - \\mat{I}\\right\\|_F^2\n\\leq \\frac{\\gamma^2}{R^{2(N-1)}} \\cdot \\|\\mat{Q}_P\\|_F^4 = \\gamma^2.\n\\]\nTherefore, \n\\[\n\\left\\|\\mat{Q}_P^T\\mat{S}^T \\mat{S}\\mat{Q}_P - \\mat{I}\\right\\|_2 \\leq \n\\left\\|\\mat{Q}_P^T\\mat{S}^T \\mat{S}\\mat{Q}_P - \\mat{I}\\right\\|_F \\leq \\gamma,\n\\]\nwhich means the singular values of $\\mat{SQ}_P$ satisfy $1-\\gamma\\leq \\sigma^2 \\leq 1+\\gamma$.\n\\end{proof}\n\nThe previous two lemmas can be combined to demonstrate that the TensorSketch matrix provides an accurate sketch within our analytical framework.\n\n\\begin{lemma}[$(1\/2, \\delta, \\epsilon)$-accurate TensorSketch Matrix]\\label{lem:structure_tensorsketch}\nGiven the sketch size,\n\\[m= \\bigO{(R^{(N-1)} \\cdot 3^{N-1})\/\\delta \\cdot (R^{(N-1)}  + 1\/\\epsilon^2)},\\]\nan order $N-1$ TensorSketch matrix $\\mat{S}\\in\\BB{R}^{m \\times s^{N-1}}$ is a $(1\/2, \\delta, \\epsilon)$-accurate sketching matrix \nfor any full rank matrix\n$\\mat{P}\\in\\BB{R}^{s^{N-1}\\times R^{N-1}}$.\n\\end{lemma}\n\\begin{proof}\nBased on \\cref{lam:singular_tensorsketch} with $\\gamma=1\/2$, for \n\\[m \\geq R^{2(N-1)}(2 + 3^{N-1})\/(1\/4 \\cdot \\delta\/2) = \\bigO{(R^{2(N-1)} \\cdot 3^{N-1})\/\\delta},\\] \n\\eqref{eq:structure1} in \\cref{def:accurate_skeching} will hold.\nBased on \\cref{lam:matmul_tensorsketch}, for $m\\geq R^{N-1}(2 + 3^{N-1})\/(\\epsilon^2\\delta)$, \n\\[\n\\|\\mat{Q}_P^T\\mat{S}^T \\mat{SB} - \\mat{Q}_P^T\\mat{B}\\|_F^2\\leq \\frac{\\epsilon^2}{R^{N-1}} \\cdot \\|\\mat{Q}_P\\|_F^2\\cdot \\|\\mat{B}\\|_F^2 = \\epsilon^2\\|\\mat{B}\\|_F^2,\n\\]\nthus \\eqref{eq:structure2} in \\cref{def:accurate_skeching} will hold. Therefore, we need\n\\begin{align*}\n m &= \\bigO{(R^{2(N-1)} \\cdot 3^{N-1})\/\\delta + (R^{(N-1)} \\cdot 3^{N-1})\/(\\epsilon^2\\delta)} \\\\\n&= \\bigO{(R^{(N-1)} \\cdot 3^{N-1})\/\\delta \\cdot (R^{(N-1)}  + 1\/\\epsilon^2)}.   \n\\end{align*}\n\\end{proof}\n\nUsing \\cref{lem:structure_tensorsketch}, we can then easily derive the upper bounds for both unconstrained and rank-constrained linear least squares with TensorSketch.\n\\begin{theorem}[TensorSketch for Unconstrained Linear Least Squares]\n\\label{thm:tensorsketch-ucls}\nGiven a full-rank matrix $\\mat{P}\\in\\BB{R}^{s^{N-1}\\times R^{N-1}}$ with $s>R$, and $\\mat{B}\\in \\mathbb{R}^{s^{N-1}\\times n}$.\nLet $\\mat{S}\\in\\BB{R}^{m \\times s^{N-1}}$ be an order $N-1$ TensorSketch matrix.\nLet $\\mat{\\widetilde{X}}_{\\text{opt}}= \\arg\\min_{\\mat{X}} \\fnrm{\\mat{SPX}-\\mat{SB}}$ and $\\mat{X}_{\\text{opt}} = \\arg\\min_{\\mat{X}} \\left\\|\\mat{PX}-\\mat{B}\\right\\|_F$. \nWith \n\\begin{equation}\\label{eq:size_tensorsketch_ucls}\n  m = \\bigO{(R^{(N-1)} \\cdot 3^{N-1})\/\\delta \\cdot (R^{(N-1)}  + 1\/\\epsilon)},  \n\\end{equation}\nthe approximation,\n$\n\\left\\|\\mat{A}\\mat{\\widetilde{X}}_{\\text{opt}}- \\mat{B}\\right\\|_F^2\t \\leq \\left(1+\\bigO{\\epsilon}\\right) \\Big\\|\\mat{A}\\mat{X}_{\\text{opt}}- \\mat{B}\\Big\\|_F^2,\n$\nholds with probability at least $1-\\delta$.\n\\end{theorem}\n\\begin{proof}\nBased on \\cref{thm:structure-ls}, to prove this theorem, we derive the sketch size $m$ sufficient to make the sketching matrix $(1\/2, \\delta, \\sqrt{\\epsilon})$-accurate. According to \\cref{lem:structure_tensorsketch}, the sketch size \\eqref{eq:size_tensorsketch_ucls} is sufficient for being $(1\/2, \\delta, \\sqrt{\\epsilon})$-accurate.\n\\end{proof}\n\n\\begin{proof}[Proof of \\cref{thm:tensorsketch-rcls}]\nBased on \\cref{thm:structure-rcls}, to prove this theorem, we derive the sketch size $m$ sufficient to make the sketching matrix $(1\/2, \\delta, \\epsilon)$-accurate. \nAccording to \\cref{lem:structure_tensorsketch}, the sketch size\n \\[m= \\bigO{(R^{(N-1)} \\cdot 3^{N-1})\/\\delta \\cdot (R^{(N-1)}  + 1\/\\epsilon^2)}\\] is sufficient for being $(1\/2, \\delta, \\epsilon)$-accurate.\n\\end{proof}\n\n\n\n\\section{Introduction}\n\\input{material_arxiv\/intro_arxiv}\n\n\\section{Background}\n\\label{sec:background}\n\\input{material_arxiv\/bg_arxiv}\n\n\\section{Background on Sketching}\n\\label{sec:sketch}\n\\input{contents\/sketching}\n\n\\section{Sketched Rank-constrained Linear Least Squares for Tucker-ALS}\n\\label{sec:sketch-rcls}\n\\input{contents\/ls}\n\n\\section{Initialization of Factor Matrices via Randomized Range Finder}\n\\label{sec:init}\n\\input{contents\/init}\n\n\\section{Cost Analysis}\n\\label{sec:cost}\n\\input{material_arxiv\/cost_arxiv}\n\n\\section{Experiments}\n\\label{sec:exp}\n\\input{material_arxiv\/exp_arxiv}\n\n\\section{Conclusions}\n\\label{sec:conclu}\n\\input{contents\/conclusion}\n\n\n\n\\small\n\\bibliographystyle{abbrv}\n\n\\subsection{Tucker Decomposition with ALS}\n\\label{sec:bg-tucker}\n\\input{contents\/bg\/bg_tucker}\n\\input{contents\/bg\/bg_tucker_alg}\n\\subsection{CP Decomposition with ALS}\n\\label{sec:bg-cp}\n\\input{contents\/bg\/bg_cp}\n\\input{contents\/bg\/bg_cp_alg}\n\\input{contents\/bg\/bg_previous}\n\\subsection{Cost Analysis for Sketched Tucker-ALS}\n\\label{subsec:cost}\n\\input{contents\/cost\/main_alg}\n\\input{contents\/cost\/cost}\n\\subsection{Cost Analysis for CP Decomposition}\n\\label{sec:cpd}\n\\input{contents\/cost\/cpd}\n\\subsection{Experiments for Tucker Decomposition}\n\\label{subsec:exp_tucker}\n\nWe compare five different algorithms for Tucker decomposition.\nTwo baselines from previous work are considered, standard HOOI and the original TensorSketch-based randomized Tucker-ALS algorithm, which optimizes only one factor in Tucker decomposition at a time~\\cite{malik2018low}.\nWe compare these to\nour new randomized algorithm (\\cref{tucker-rals}) based on TensorSketch, random leverage score sampling, and deterministic leverage score sampling. For each randomized algorithm, we test both random initialization for factor matrices as well as the initialization scheme based on RRF detailed in \\cref{sec:init}. For the baseline HOOI algorithm, we report the performance with both random and HOSVD initializations. \nWe use four synthetic tensors to evaluate these algorithms.\n\\begin{enumerate}[itemsep=0pt,leftmargin=*]\n    \\item \\label{tsr:dense} \\textbf{Dense tensors with specific Tucker rank}. We create order 3 tensors based on randomly-generated factor matrices $\\mat{B}^{(n)}\\in \\BB{R}^{s\\times R_{\\text{true}}}$ and a core tensor $\\tsr{C}$,\n    \\begin{equation}\\label{eq:tsr_dense}\n     \\tsr{T} = \\tsr{C}\\times_1\\mat{B}^{(1)}\\times_2\\mat{B}^{(2)}\\times_3\\mat{B}^{(3)}. \n    \\end{equation}\n    Each element in the core tensor and the factor matrices are i.i.d. normally distributed random variables $\\mathcal{N}(0,1)$. \n    The ratio $R_{\\text{true}}\/R$, where $R$ is the decomposition rank, is denoted as $\\alpha$. \n    \\item \\label{tsr:dense-bias} \\textbf{Dense tensors with strong low-rank signal}. We also test on dense tensors with strong low-rank signal,\n    \\begin{equation}\n        \\tsr{T}^{(b)} = \\tsr{T} + \\sum_{i=1}^n \\lambda_i \\vcr{a}^{(1)}_i \\circ \\vcr{a}^{(2)}_i \\circ \\vcr{a}^{(3)}_i. \n    \\end{equation}\n    $\\tsr{T}$ is generated based on \\eqref{eq:tsr_dense}, and each vector $\\vcr{a}_i^{(j)}$ has unit 2-norm. The magnitudes $\\lambda_i$ for $i\\in[n]$ are constructed based on the power-law distribution,\n    $\\lambda_i = C \\frac{\\|\\tsr{T}\\|_F}{i^{1+\\eta}}$. In our experiments, we set $n=5,C=3$ and $\\eta=0.5$. This tensor is used to model data whose leading low-rank components obey the power-law distribution, which is common in real datasets.\n    \\item \\label{tsr:sparse} \\textbf{Sparse tensors with specific Tucker rank}. We generate tensors based on \\eqref{eq:tsr_dense} with each element in the core tensor and factor matrices being\n    an i.i.d normally distributed random variable $\\mathcal{N}(0,1)$ with probability $p$ and zero otherwise. \n    Since each element,\n    \\begin{equation}\\label{eq:tsr_sparse}\n    \\tsr{T}(i,j,k) = \\sum_{x,y,z} \\mat{B}^{(1)}(i, x)\\cdot \\mat{B}^{(2)}(j, y)\\cdot \\mat{B}^{(3)}(k, z)\\cdot \\tsr{C}(x,y,z), \n    \\end{equation}\n    and \n    \\[\n    P\\left[\n    \\mat{B}^{(1)}(i, x)\\cdot\\mat{B}^{(2)}(j, y)\\cdot\\mat{B}^{(3)}(k, z)\\cdot\\tsr{C}(x,y,z) \\neq 0\\right]\n    = p^4,\n    \\]\n    the expected sparsity of $\\tsr{T}$, which is equivalent to the probability that each element $\\tsr{T}(i,j,k)=0$, is lower-bounded by $1 - R_{\\text{true}}^3p^4$. \n    Through varying $p$, we generate tensors with different expected sparsity. \n\n    \\item \\label{tsr:sparse-bias} \\textbf{Tensors with large coherence}. \n    We also test on tensors with large coherence,\n    \\begin{equation}\n        \\tsr{T}^{(b)} = \\tsr{T} + \\tsr{N}. \n    \\end{equation}\n    $\\tsr{T}$ is generated based on \\eqref{eq:tsr_dense} or \\eqref{eq:tsr_sparse}, and $\\tsr{N}$ contains $n \\ll s$ elements with random positions and same large magnitude. In our experiments, we set $n=10$, and each nonzero element in $\\tsr{N}$ has the i.i.d. normal distribution $\\mathcal{N}(\\|\\tsr{T}\\|_F\/\\sqrt{n}, 1)$, which means the expected norm ratio $\\E[\\|\\tsr{N}\\|_F\/\\|\\tsr{T}\\|_F] = 1$. This tensor has large coherence and is used to test the robustness problem detailed in \\cref{sec:init}.\n\\end{enumerate}\n\nFor all the experiments, we run 5 ALS sweeps  unless otherwise specified, and calculate the fitness based on the output factor matrices as well as the core tensor. We observe that 5 sweeps are sufficient for both HOOI and randomized algorithms to converge. For each randomized algorithm, we set the sketch size to be $KR^2$. \nFor the RRF-based initialization, we set the sketch size ($k$ in \\cref{alg:rrf}) as $\\sqrt{K}R$.\n\n\n\n\\begin{figure}[!ht]   \n\\centering\n\\subfloat[Tensor \\ref{tsr:dense} with $s=200$]{\\includegraphics[width=0.33\\textwidth, keepaspectratio]{figs\/fitness-sweep-tucker-tensor1.pdf}\\label{figs-tuckerdetail1}}\n\\subfloat[Tensor \\ref{tsr:dense-bias} with $s=200$]{\\includegraphics[width=0.33\\textwidth, keepaspectratio]{figs\/fitness-sweep-tucker-tensor2.pdf}\\label{figs-tuckerdetail2}}\n\\subfloat[Tensor \\ref{tsr:sparse-bias} with $s=1000$]{\\includegraphics[width=0.33\\textwidth, keepaspectratio]{figs\/fitness-sweep-tucker-tensor3.pdf}\\label{figs-tuckerdetail3}}\n\n\\caption{Detailed fitness-sweeps relation for Tucker decomposition of three dense tensors with different parameters. \nFor Tensor \\ref{tsr:sparse-bias}, $\\tsr{T}$ is generated based on \\eqref{eq:tsr_dense}.\nFor all the experiments, we set $R=5, \\alpha=1.6$, and $K=16$. \nIn the plots, Lev, Lev-fix, and TS denote our new sketched Tucker-ALS scheme with leverage score random sampling, leverage score deterministic sampling, and TensorSketch, respectively. TS-ref denotes the reference sketched Tucker-ALS algorithm with TensorSketch.\nHOOI is initialized with HOSVD, and all other methods are initialized with RRF (\\cref{alg:rrf}). Markers represent the results per sweep.\n}\n\\label{fig:tucker-fitness-sweep}\n\\end{figure}\n\n\\begin{figure}[!ht]   \n\\centering\n\\subfloat[Tensor \\ref{tsr:dense} with $\\alpha=1.2, K=16$]{\\includegraphics[width=0.50\\textwidth, keepaspectratio]{figs\/dense-epsilon0.5-R5-gaussian.pdf}\\label{figs-dense1}}\n\\subfloat[Tensor \\ref{tsr:dense} with $\\alpha=1.2$ and HOSVD\/RRF init]{\\includegraphics[width=0.50\\textwidth, keepaspectratio]{figs\/dense-varyepsilon-R5-gaussian.pdf}\\label{figs-dense2}}\n\n\\subfloat[Tensor \\ref{tsr:dense} with $\\alpha=1.6, K=16$]{\\includegraphics[width=0.50\\textwidth, keepaspectratio]{figs\/dense-epsilon0.5-R5-alpha1.6-gaussian.pdf}\\label{figs-dense3}}\n\\subfloat[Tensor \\ref{tsr:dense} with $\\alpha=1.6$ and HOSVD\/RRF init]{\\includegraphics[width=0.50\\textwidth, keepaspectratio]{figs\/dense-varyepsilon-R5-alpha1.6-gaussian.pdf}\\label{figs-dense4}}\n\n\\caption[]{Experimental results for Tucker decomposition of Tensor \\ref{tsr:dense}. For all the experiments, we set $s=200$ and $R=5$. \nHOSVD\/RRF means HOOI is initialized with HOSVD, and all other methods are initialized with RRF (\\cref{alg:rrf}).\n\\textbf{(a)(c)} Box plots of the final fitness for each algorithm with different input tensors. Each box is based on 10 experiments with different random seeds. \nEach box shows the 25th-75th quartiles, the median is indicated by a horizontal line inside the box, and outliers are displayed as dots.\n\\textbf{(b)(d)} Relation between the final fitness and sketch size parameter $K$ for each algorithm with different input tensors. Each data point is the mean of 10 experimental results with different random seeds. \n}\n\\label{fig:dense}\n\\end{figure}\n\n\n\\begin{figure}[!ht]   \n\\centering\n\\subfloat[Tensor \\ref{tsr:dense-bias} with $\\alpha=1.6, K=16,s=200$]{\\includegraphics[width=0.50\\textwidth, keepaspectratio]{figs\/dense-bias-epsilon0.5-R5-alpha1.6-gaussian.pdf}\\label{figs-dense5}}\n\\subfloat[Tensor \\ref{tsr:dense-bias} with $\\alpha=1.6, s=200$ and HOSVD\/RRF init]{\\includegraphics[width=0.50\\textwidth, keepaspectratio]{figs\/dense-bias-varyepsilon-R5-alpha1.6-gaussian.pdf}\\label{figs-dense6}}\n\n\\subfloat[Tensor \\ref{tsr:sparse-bias} with $\\alpha=1.6, K=16$, $s=1000$]{\\includegraphics[width=0.50\\textwidth, keepaspectratio]{figs\/dense-coherence-epsilon0.25-R5-s1000-alpha1.6-gaussian.pdf}\\label{figs-dense7}}\n\\subfloat[Tensor \\ref{tsr:sparse-bias} with $\\alpha=1.6, s=1000$ and HOSVD\/RRF init]{\\includegraphics[width=0.50\\textwidth, keepaspectratio]{figs\/dense-coherence-varyepsilon-R5-alpha1.6-s1000-gaussian.pdf}\\label{figs-dense8}}\n\n\\caption[]{Experimental results for Tucker decomposition of dense tensors. \nFor Tensor \\ref{tsr:sparse-bias}, $\\tsr{T}$ is generated based on \\eqref{eq:tsr_dense}.\nFor all the experiments, we set $R=5$. \n\\textbf{(a)(c)} Box plots of the final fitness for each algorithm with different input tensors. Each box is based on 10 experiments with different random seeds. \n\\textbf{(b)(d)} Relation between the final fitness and sketch size parameter $K$ for each algorithm with different input tensors. \n}\n\\label{fig:dense-bias}\n\\end{figure}\n\n\n\n\n\\subsubsection{Experimental Results for Dense Tensors}\n\nWe show the detailed fitness-sweeps relation for different dense tensors in \\cref{fig:tucker-fitness-sweep}. The reference randomized algorithm suffers from unstable convergence as well as low fitness, while our new randomized ALS scheme, with either leverage score sampling or TensorSketch, converges faster than the reference randomized algorithm and reaches higher accuracy. \n\nWe show the final fitness for each method on different dense tensors in  \\cref{fig:dense} and \\cref{fig:dense-bias}. For Tensor \\ref{tsr:dense}, we test on two different values of the true rank versus decomposition rank ratio, $\\alpha$. \nWe do not report the results for $\\alpha=1$, which corresponds to the exact Tucker decomposition, since all the randomized algorithms can find the exact decomposition easily.\nWe also test on Tensor \\ref{tsr:dense-bias} and Tensor \\ref{tsr:sparse-bias}. As can be seen in the figure, our new randomized ALS scheme, with either leverage score sampling or TensorSketch, outperforms the reference randomized algorithm for all the tensors. The relative fitness improvement ranges from $4.5\\%$ (\\cref{figs-dense5},\\ref{figs-dense6}) to $22.0\\%$ (\\cref{figs-dense3},\\ref{figs-dense4}) when $K=16$.\n\n\\cref{figs-dense2},\\ref{figs-dense4},\\ref{figs-dense6},\\ref{figs-dense8} show the relation between the final fitness and $K$. As is expected, increasing $K$ can increase the accuracy of the randomized linear least squares solve, thus improving the final fitness.  With our new randomized scheme, the relative final fitness difference between HOOI and the randomized algorithms is less than  $8.5\\%$ when $K=16$, indicating the efficacy of our new scheme. \n\n\\cref{figs-dense1},\\ref{figs-dense3},\\ref{figs-dense5},\\ref{figs-dense7} include a comparison between random initialization and the initialization scheme based on RRF detailed in \\cref{alg:rrf}. For Tensor \\ref{tsr:dense}, both initialization schemes have similar performance. For the deterministic leverage score  sampling on Tensor \\ref{tsr:dense-bias} (\\cref{figs-dense5}), using RRF-based initialization substantially decreases variability of attained accuracy. For leverage score sampling on Tensor \\ref{tsr:sparse-bias} (\\cref{figs-dense7}), we observe that the random initialization is not effective, resulting in approximately zero final fitness. \nThis is because the random initializations are far from the accurate solutions, and some elements with large amplitudes are not sampled in all the ALS sweeps, consistent with our discussion in \\cref{sec:init}. With the RRF-based initialization, the output fitness of the algorithms based on leverage score sampling is close to HOOI. Therefore, our proposed initialization scheme is important for improving the robustness of leverage score sampling.\n\nAlthough leverage score sampling has a better sketch size upper bound, based on analysis in \\cref{sec:sketch-rcls}, the random leverage score sampling scheme performs similar to TensorSketch for the tested dense tensors. For leverage score sampling, \\cref{figs-dense6},\\ref{figs-dense8} shows that when the sketch size is small ($K=4$), the deterministic leverage score sampling scheme outperforms the random sampling scheme for Tensor \\ref{tsr:dense-bias} and Tensor \\ref{tsr:sparse-bias}. This means that when the tensor has a strong low-rank signal, the deterministic sampling scheme can be better, consistent with the results in \\cite{papailiopoulos2014provable}. \n\n\n\\subsubsection{Experimental Results for Sparse Tensors}\n\\input{material_arxiv\/exp_sparse_tucker}\n\n\n\\subsection{Experiments for CP Decomposition}\n\n\\begin{figure}[!ht]   \n\\centering\n\\subfloat[$\\alpha=1.2$]{\\includegraphics[width=0.5\\textwidth, keepaspectratio]{figs\/s2000-epsilon0.25-R10-alpha1.2-iter25-CP.pdf}\\label{figs-cp1}}\n\\subfloat[$\\alpha=1.6$]{\\includegraphics[width=0.5\\textwidth, keepaspectratio]{figs\/s2000-epsilon0.25-R10-alpha1.6-iter25-CP.pdf}\\label{figs-cp2}}\n\n\\caption{Relation between final fitness and sparsity parameter $p$ for CP decomposition. For all the experiments, we set $s=2000, R=10$ and $K=16$. In the plots, CP denotes running CP-ALS, Tucker+CP denotes running the Tucker HOOI + CP-ALS algorithm, Lev CP denotes running leverage score sampling based randomized CP-ALS, and Lev Tucker+CP denotes running the leverage score sampling based Tucker-ALS + CP-ALS algorithm.\nEach box is based on 10 experiments with different random seeds. \n}\n\\label{fig:cp}\n\\end{figure}\n\n\\begin{figure}[!ht]   \n\\centering\n\\subfloat[$p=0.5$]{\\includegraphics[width=0.33\\textwidth, keepaspectratio]{figs\/fitness-sweep-cp-1.pdf}\\label{figs-cp3}}\n\\subfloat[$p=0.1$]{\\includegraphics[width=0.33\\textwidth, keepaspectratio]{figs\/fitness-sweep-cp-2-p-0.1.pdf}\\label{figs-cp4}}\n\\subfloat[$p=0.02$]{\\includegraphics[width=0.33\\textwidth, keepaspectratio]{figs\/fitness-sweep-cp-3-p-0.02.pdf}\\label{figs-cp5}}\n\n\\caption{Detailed fitness-sweeps relation for CP decomposition of three tensors with different parameters. For all the experiments, we set $s=2000, R=10, \\alpha=1.2$ and $K=16$. Markers represent the results per sweep. For Tucker + CP algorithms, the fitness shown for $i$th sweep is the output fitness after running $i$ Tucker sweeps along with 25 CP-ALS sweeps on core tensors afterwards.\n}\n\\label{fig:cp-fitness-sweep}\n\\end{figure}\n\nWe show the efficacy of accelerating CP decomposition via performing Tucker compression first. We compare four different algorithms, the standard CP-ALS algorithm, the Tucker HOOI + CP-ALS algorithm, sketched CP-ALS, where the sketching matrix is applied to each linear least squares subproblem \\eqref{eq:subproblem}, as well as the sketched Tucker-ALS + CP-ALS algorithm shown in \\cref{alg:cp-rand-tucker}. \nRandom leverage score sampling is used for sketching, since it has been shown to be efficient for both Tucker (\\cref{subsec:exp_tucker}) and CP (reference \\cite{larsen2020practical}) decompositions. We use the following synthetic tensor to evaluate these four algorithms,\n\\begin{equation}\n    \\tsr{T} = \\sum_{i=1}^{R_{\\text{true}}} \\vcr{a}^{(1)}_i \\circ \\vcr{a}^{(2)}_i \\circ \\vcr{a}^{(3)}_i, \n\\end{equation}\nwhere each element in $\\vcr{a}^{(j)}_i$ is an i.i.d normally distributed random variable $\\mathcal{N}(0,1)$ with probability $p$ and is zero otherwise. \nThis guarantees that the expected sparsity of $\\tsr{T}$ is lower-bounded by $1 - R_{\\text{true}}p^3$. \nThe ratio $R_{\\text{true}}\/R$, where $R$ is the decomposition rank, is denoted as $\\alpha$. \n\nFor (sketched) Tucker + CP algorithms, we run 5 (sketched) Tucker-ALS sweeps first, and then run the CP-ALS algorithm on the core tensor for 25 sweeps. RRF-based initialization is used for Tucker-ALS, and HOSVD on the core tensor is used to initialize the factor matrices of the small CP decomposition problem. For (sketched) CP-ALS algorithms, we also use the RRF-based initialization and run 30 sweeps afterwards, which is sufficient for CP-ALS to converge based on our experiments. This initialization makes sure that leverage score sampling is effective for sparse tensors. We set the sketch size as $KR^2$ for  both algorithms. The constant factor $K$ reveals the accuracy of each subproblem. \nFor the RRF-based initialization, we set the sketch size ($k$ in \\cref{alg:rrf}) as $\\sqrt{K}R$.\n\nWe show the relation between final fitness and the tensor sparsity parameter, $p$, in \\cref{fig:cp}. As can be seen, for all the tested tensors, the Tucker + CP algorithms perform similarly, and usually better than directly performing CP decomposition. When the input tensor is sparse ($p=0.1$ and $0.02$), the advantage of the Tucker + CP algorithms is greater.\nThe sketched Tucker-ALS + CP-ALS scheme has a comparable performance compared to Tucker HOOI + CP-ALS, while requiring less computation.\n\nWe show the detailed fitness-sweeps relation in \\cref{fig:cp-fitness-sweep}. We observe that for (sketched) CP-ALS, more than 10 sweeps are necessary for the algorithms to converge. On the contrary, less than 5 Tucker-ALS sweeps are needed for the sketched Tucker + CP scheme, making it more efficient. \n\nIn summary, we observe CP decomposition can be more efficiently and accurately calculated based on the sketched Tucker + CP method.\n\n\n\\subsection{Organization}\n\\cref{sec:background} introduces notation used throughout the paper, ALS algorithms for Tucker and CP decompositions, and previous work. \\cref{sec:sketch} introduces TensorSketch and leverage score sampling techniques. In \\cref{sec:sketch-rcls}, we provide sketch size upper bounds for the  rank-constrained linear least squares subproblems in sketched Tucker-ALS. Detailed proofs are presented in \\cref{appendix:detailed_proofs}. In \\cref{sec:init}, we introduce the initialization scheme for factor matrices via randomized range finder. In \\cref{sec:cost}, we present detailed cost analysis for our new sketched ALS algorithms. We present experimental results in \\cref{sec:exp} and conclude in \\cref{sec:conclu}.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{APPENDIX}\n\n\n\n\n\\keywords{hls4ml, machine learning, neural networks, quantum control, FPGA, low-power, low-latency}\n\n\n\\maketitle\n\n\n\n\n\n\\section{Introduction}\n\\label{sec:intro}\n\n\nQuantum computing has great potential but today's quantum computers are still encumbered by errors that limit their practical usefulness~\\cite{fellous20211, ball20211}. Quantum control seeks to enable  precise manipulation of quantum hardware via control-theoretical approaches~\\cite{ozguler2022numerical, koch2022quantum, ozguler2022dynamics}. The result would be an increase in performance of the imperfect hardware. It is desirable to apply such techniques in actual quantum-computational environments, necessitating purpose-built control hardware that can function at low temperature~\\cite{butko20201, xue2021cmos, koolstra2022monitoring, stefanazzi2022qick, alam2022quantum}. By approximating existing quantum control algorithms using machine learning (ML), we hope to achieve comparable functionality using significantly less computation. The resulting model could then be implemented as a hardware accelerator at reasonable costs~\\cite{overwater2022neural}.\n\n\n\nA qubit (quantum bit) is a quantum computing system's smallest unit of information. Its quantum state can be described as a linear combination of two orthonormal states $\\left|0\\right\\rangle$ and $\\left|1\\right\\rangle$: $A$ = $x$ $\\left|0\\right\\rangle$ + $y$ $\\left|1\\right\\rangle$, where $x$ and $y$ are complex numbers. Quantum gates apply specified transformations, changing  quantum states accordingly. We desire to create a target gate $U$ which optimally changes quantum state $A$ into $U*A$. Time-varying pulses are used to produce such action. For some gate, $U(\\beta)$, where $\\beta$ is a gate parameter, the problem of gate synthesis can be understood as finding optimal pulse parameters $\\alpha$ to produce action $U(\\beta)$. Devising pulses for a given value of $\\beta$ is computationally expensive because the size of the underlying pulse parameters (the vector $\\alpha$) can be quite large. Furthermore, there are infinitely many possible $\\beta$ (as generally $\\beta$ is selected from some subset of $\\mathbb{R}$). Conventional tools for such calculations include Juqbox.jl, Quandary, and QuTiP~\\cite{juqboxjl, quandary, qutip}. These tools utilize optimization techniques such as gradient ascent to discover optimal $\\alpha$.\n\nIn this work, a machine learning model will be trained to determine the appropriate $\\alpha$ for producing $U(\\beta)$ for any $\\beta$ (Fig.~\\ref{fig:overview}). The model will be implemented as a hardware accelerator via {\\tt{hls4ml}}~\\cite{hls4ml}. hls4ml is a toolset that converts pre-trained ML models into accelerators for FPGA or ASIC implementation with low power or ultra-low latency goals. It requires little designer overhead. Furthermore, the integration with the Google QKeras library allows aggressively quantized deep neural networks to be implemented. The final model is highly resource-efficient and sacrifices little to no accuracy. \nSince the goal of our present work is to produce an accelerator capable of operating within the harsh conditions of a quantum computing environment, it is desirable for such an accelerator to be small to minimize energy dissipation within the low-temperature quantum environments managed by systems of limited cooling capacity. Hardware implementations  need to be fast so that quantum computations can be done within the coherence time since noise is a big bottleneck for current quantum devices.\n\n\n\nNote that hardware implementations that function as lookup tables mapping from a fixed number of $\\beta$ to the corresponding $\\alpha$ do exist \\cite{liang20221}. However, such tables are limited by their precision, as only entries for a finite number of $\\beta$ can be stored. Further discussion of the advantages and disadvantages of such systems can be found in Section~\\ref{sec:lut}.\n \n \nFor this exploration, a simplified scenario is specified as follows: The gate $U(\\beta)$ (X-gate) rotates a qubit about the x-axis by angle $-\\pi{}\\leq{}\\beta{}\\leq{}\\pi{}$. We model the qubit as the ground and first excited states of a transmon, a type of superconducting qubit. The pulses are specified by a 20-dimensional vector $\\alpha$. The dimension \"20\" is due to the B-spline parametrization we follow in Juqbox \\cite{juqboxjl}, which is based on the numerical techniques in Refs. \\cite{petersson2020discrete, petersson2021optimal}. The dimension of the vector $\\alpha$ is proportional to the multiplication of number of carrier frequencies and number of B-splines (here, 1 and 10, respectively). X-gate brings the state (ground state) to a superposition state as shown in Figure~\\ref{fig:transformsample}, which visualizes on a Bloch sphere the qubit being rotated from its initial $\\left|0\\right\\rangle$ quantum state.\n\n\n\\begin{figure}[t]\n\\centering\n\\captionsetup{justification=centering, format=hang}\n\\includegraphics[width=0.95\\columnwidth]{fig\/esp_overview.pdf}\n\\caption{Simplified overview.}\n\\label{fig:overview}\n\\end{figure}\n\n\\begin{figure}[t]\n\\centering\n\\captionsetup{justification=centering, format=hang}\n\\includegraphics[width=0.95\\columnwidth]{fig\/esp_transformsample.pdf}\n\\caption{Bloch sphere visualization of a rotation transformation starting from the $\\left|0\\right\\rangle$ quantum state.}\n\\label{fig:transformsample}\n\\end{figure}\n\n\nThe accelerator will be evaluated for accuracy, cost, and performance. Accuracy is measured by comparing the ML-model results to those of conventional quantum control algorithms. This metric is considered both for the software model and its synthesized hardware implementation, which may perform differently due to quantization. Accuracy evaluation includes both traditional ML measures such as mean squared error (MSE) and domain-specific measures such as quantum gate fidelity. Cost is measured as the FPGA area of the synthesized result. Performance is measured via an analysis of accelerator latency and throughput.\n\nFidelity is a metric of the overlap between two unitaries (gates), $U_1$ and $U_2$. It can be defined in several ways. We use the definition \\cite{pedersen20071}:\n\\begin{equation}\nF= \\displaystyle \\frac{M+\\left|tr({U})\\right|^2}{M(M+1)}\\label{eq:QuantumFidelity},\n\\end{equation} \nwhere $U = U_1^\\dagger U_2$ and $M=2$ is the dimensionality of the Hilbert space for single qubit. We use the following definitions for three types of unitaries considered in this paper. The ``Golden'' gate is the target gate. \"Optimized\" gate is calculated via pulse optimizer (\\texttt{Juqbox.jl} \\cite{juqboxjl} for this paper), which starts from a random pulse and finds the optimal pulse until reaching a target fidelity (calculated as overlap between \"optimized\" and \"golden\" gates). The ``Predicted'' gate is the one calculated from the ML model. We check the overlap between these three types of gates, where $U_1$ and $U_2$ are two of these three types (golden, optimized, predicted). \n\n\n\\section{Model development}\n\\label{sec:model_development}\n\nOur initial dataset consists of 101 samples. Each sample is a rotation angle upon the x-axis of a qubit ($\\beta$ value) and the corresponding quantum state transformations in the form of 20 parameters ($\\alpha$ vector) that were generated with Juqbox, i.e., an open-source software package designed to solve quantum optimal control problems in closed systems~\\cite{juqboxjl}. Our later analysis has shown that the models perform well for most angles other than $-\\pi$. This was because the dataset used the same pulses for the angles $-\\pi$ and $\\pi$, as both transformations produce the same final quantum state (i.e., the transformation for $-\\pi$ actually is carried out in the positive direction). Thus we decided to remove from the dataset the samples that were associated with $-\\pi$. We used a 60-20-20-percent split for training, validation, and test sets.\n\nAs a starting point in our neural-architecture exploration, we chose the multi-layer perceptron (MLP) shown in Fig.~\\ref{fig:mlp_forward}. The model has a total of seven hidden layers and 1,040 trainable parameters. We trained the model in Tensorflow with Adam optimizer and minimized the mean-squared error (MSE) as a loss function. We set a target of $5,000$ epochs with an early-stopping callback~\\cite{earlystopping}.\nThe final MSE value that we measured on the test set was $7.908\\times{}10^{-8}$. Since there is no ``correct'' value for MSE, but the lower, the better, and a zero value means the model is perfect, we assumed this value as our reference for the rest of the neural-architecture exploration.\n\n\nWe also compared the expected and predicted results of the pulses on the Bloch sphere using the graphical features of the QuTiP framework~\\cite{qutip}. Fig.~\\ref{fig:transformcomparison} shows the similarities of the results. This visual comparison further proves that the proposed model could learn the relationship between the final angle and the required quantum state transformations.\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=0.95\\columnwidth]{fig\/model_forward_notext.pdf}\n\\caption{The initial model architecture: a multi-layer perceptron with seven hidden layers, 1,040 parameters.\\label{fig:mlp_forward}}\n\\end{figure}\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.95\\columnwidth]{fig\/bloch_sphere_comparison.pdf}\n\\caption{Comparison of transformations produced by Juqbox optimizer and our machine-learning model. Each row illustrates results for a different angle $\\beta{}$.}\n\\label{fig:transformcomparison}\n\\end{figure}\n\nWe adopted the gate fidelity in Eq.~\\ref{eq:QuantumFidelity} as our last tool to quantify the quality of our results further. Figure~\\ref{fig:fidelity} shows the fidelity when comparing the gates produced by our model, the traditional software optimizer, and mathematically-derived ground truth. Again, it is evident that the ML model performs poorly around $\\beta{}=-\\pi$, suggesting that the $-\\pi$ row of the training dataset be removed so that the sign of $\\beta$ values is consistent with the direction of the resultant rotation transformation.\n\nFigure~\\ref{fig:fidelity2} shows the same results with the first three $\\beta{}$ removed. The traditional software optimizer was configured to ensure fidelity greater than 0.999 between its results and the ground truth, as visible in the figure. It is interesting to note how the curve indicating overlap between the ML model outputs and the ground truth appears as a ``smoothed'' version of the overlap between the ML model outputs and the optimizer. Such a result suggests that the ML model can ignore lots of the random noise from the optimizer while effectively learning how to produce the target gate. The ML model produces gates with a gate fidelity of over $0.99$ for all remaining $\\beta$, suggesting practical usefulness.\n\n\n\\begin{figure}[t]\n\\centering\n\\captionsetup{justification=centering, format=hang}\n\\includegraphics[width=0.95\\columnwidth]{fig\/fidelity1.pdf}\n\\caption{Gate fidelity of forward model. Gates from ML model (``Predicted''), Juqbox optimizer (``Optimized''), and golden mathematically-derived (``Golden'') gates.}\n\\label{fig:fidelity}\n\\end{figure}\n\n\\begin{figure}[t]\n\\centering\n\\captionsetup{justification=centering, format=hang}\n\\includegraphics[width=0.95\\columnwidth]{fig\/fidelity2.pdf}\n\\caption{Gate fidelity of forward model. Outlier results for smallest three $\\beta$ removed.}\n\\label{fig:fidelity2}\n\\end{figure}\n\n\\section{Model codesign with hls4ml}\n\nhls4ml is an open-source framework for the codesign of optimized neural networks and deployment on field-programmable gate arrays (FPGAs) and custom hardware (ASIC)~\\cite{hls4ml}. At its core, hls4ml translates machine-learning models from common open-source software frameworks such as Tensorflow into a register-transfer level (RTL) implementation using high-level synthesis (HLS) tools~\\cite{nane2015survey}.\n\nThe hls4ml framework originates from the Fast Machine Learning for Science community~\\cite{deiana2022applications}, whose focus is the development of tools for scientific applications. The framework includes tools for the design-space exploration and the final FPGA or ASIC implementations. The resulting hardware implementations are configurable, spatial dataflow architectures tailored for speed and efficiency with extreme flexibility in the data-type precision~\\cite{hendrik2022opensource}.\n\nIn hls4ml, a designer can trade off the performance (i.e., latency and throughput) and resource utilization for a model by varying the parallelization of the algorithm via several configuration parameters. For example, the \\emph{reuse factor} (RF) parameter controls how many times each multiplier resource is used in the final hardware implementation: a designer with the goal of low latency will choose the lowest RF value.\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=0.95\\columnwidth]{fig\/model_qmodel1_notext.pdf}\n\\caption{A smaller multi-layer perceptron with six hidden layers and 783 parameters.\\label{fig:qmodel1}}\n\\end{figure}\n\nIn this work, we used hls4ml, and additionally, we combined quantization-aware training in QKeras~\\cite{qkeras} with model- and hardware-centric optimizations to find models that simultaneously accomplish the goals of high accuracy, low latency, and low FPGA-resource usage. We aimed to minimize resources such as onboard FPGA memory (BRAM), digital signal processing (arithmetic) blocks (DSPs), registers (flip-flops, or FFs), and programmable logic (lookup tables, or LUTs).\n\nOur experiments targeted three off-the-shelf FPGA-based development boards with decreasing resources, thus making the codesign effort increasingly challenging: a Digilent Genesys 2, an Avnet Ultra96-V2, and a Digilent Arty A7-100T.\n\n\\subsection{Initial hardware implementation}\nThe Digilent Genesys 2 board is a development board equipped with an AMD\/Xilinx Kintex-7 FPGA chip (\\texttt{xc7k325tffg900-2}), which comes with 890 BRAMs, 840 DSPs, 407,600 FFs, and 203,800 LUTs. We passed as input of hls4ml the model developed in Sec.~\\ref{sec:model_development} with a target clock period of 5 ns, a reuse factor of 1, and fixed-point precision with overall 16 bits for word width. The resulting implementation was a pipeline with a latency\\footnote{The latency of a pipeline is the time required for one input to pass through the system from start to end.} of 35 cycles and initiation interval\\footnote{The initiation interval of a pipeline is the time necessary for the system to process the next input (or to produce the next output).} of 1. The synthesized model used 54\\% of the available DSPs and 13\\% of available LUTs (other resources are not discussed as their utilization remains well below availability in this and all further experiments described herein).\n\n\n\n\\subsection{Shrinking the model}\\label{sec:qmodel1}\nThe Avnet Ultra96-V2 is a smaller development board equipped with an AMD\/Xilinx Zynq UltraScale MPSoC (\\texttt{xczu3eg-sbva484-1-e}). This SoC combines four ARM Cortex A53 cores and programmable logic that offers 432 BRAMs, 360 DSPs, 141,1120 FFs, and 70,560 LUTs. The synthesis of our previous model returned a DSP utilization of 220\\% and LUT utilization of 38\\%. Thus, we took additional steps to reduce the resource requirements of our hardware implementation. We removed one hidden layer and some nodes from the remaining hidden layers. The resulting model shown in Fig.~\\ref{fig:qmodel1} had a total of 783 parameters. \n\n\nTo reduce the final resource requirements on FPGA, besides reducing the overall number of model parameters, we could have increased the reuse factor in the hls4ml configuration, but we decided to keep the initial value of one. While a larger reuse factor would significantly decrease the size of the implementation, doing so also decreases parallelism, impacting latency and throughput.\n\n\nInstead, to address the resource usage, we opted for a more aggressive quantization during the quantization-aware training with QKeras. We analyzed the model inputs and outputs, and for each layer, we chose a fixed-point precision with a 2-bit-integer part and either 10- or 11-bits for the overall word width. The new model trained in QKeras has an MSE of $6.6093\\times{}10^{-8}$, which is slightly worse than the original model from Sec.\\ref{sec:model_development}. But such a change is significant for the underlying FPGA-resource mapping since multipliers on 12 and fewer bits get mapped on LUTs rather than the scarcer DSPs. The resulting model fitted excellently within the Ultra96-V2 constraints: it uses 41\\% of DSPs and 58\\% of LUTs. \n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.95\\columnwidth]{fig\/model-qmodel1-wa3-resultgraph.pdf}\n\\caption{Gate fidelity of the final implementation. Predicted-golden fidelity exceeds $0.99$ for all angles ($\\beta{}$ values).}\n\\label{fig:fidelity_qmodel1_wa3}\n\\end{figure}\n\n\n\n\\subsection{Final implementation}\nThe third board we tested is an Arty A7-100T with an AMD\/Xilinx Artix-7 chip (\\texttt{xc7a100t-csg324-1}). This chip has fewer resources but shows similarities to a future eFPGA platform of interest: 270 BRAMs, 240 DSPs, 126,800 FFs, and 63,400 LUTs.\n\n\nWe adopted the same MLP architecture as in Fig.~\\ref{fig:qmodel1}. However, we chose a mixed approach to balance DSP and LUT usage and the overall model accuracy. We used word lengths of 14, 12, 12, and 16 bits (and thus DSPs) for the first three layers and the last layer, respectively, while the word length for the remaining layers was 10 (and thus implemented in LUTs). The number of bits for the integer part was two for all layers, except the output layer, which has zero bits for the integer part.\nThe MSE measured on the test set was $5.7473\\times{}10^{-8}$.\n\nTo fit the board-resource constraints, after the quantization-aware training phase, we manually edited the hls4ml-generated design. We set a smaller 10- and 12-bit-word length for the third and output layer, respectively. Such changes decreased the resource requirements while largely preserving performance.\nThe synthesized implementation fit the Artix-7 chip with a 99\\% DSP utilization and 62\\% LUT utilization. The resulting implementation was a pipeline with an initiation interval of 1 and a latency of 35 cycles. Looking at Figure~\\ref{fig:fidelity_qmodel1_wa3}, we see that the fidelity between the gates produced by the model and the target gates is above $0.99$ for all $\\beta$.\n\n\n\\section{Further Considerations}\n\nIn this section, we discuss some additional topics and future lines of research.\n\n\\subsection{Even higher fidelity}\nOur final model achieves a gate fidelity of over $0.99$ for all $\\beta$ examined. However, it is always desirable to increase the fidelity even further. For example, our training data reaches fidelity of over $0.999$ for all $\\beta{}$, so the question arises: Would it be possible for a model trained on this data to achieve the same level of performance? Various solutions exist that can be combined with our current codesign flow based on hls4ml.\nWe can employ AutoML solutions to automatically discover a more performant model. These approaches are also known as neural architecture search (NAS)~\\cite{elsken2019neural}. Tools like AutoKeras~\\cite{autokeras} use a process of searching through neural network architectures to best address the optimal modeling task.\n\nWe have also observed that perhaps the function to compute each $\\alpha$ differs greatly across the pulse parameters. Thus, we can devise a branched model in which each output is preceded by a hidden layer of its own, with a shared hidden layer at the top connected to the input. We started investigating such a model as in Fig.~\\ref{fig:branchedmodel}, but we are still in the early phases of the analysis. \nSimilarly, we are investigating larger models (up to 20 times the current number of parameters) that would require larger FPGAs but provide one order for magnitude lower MSE and fidelity that exceeds 0.999, as shown in Fig.~\\ref{fig:fidelity_largemodel_testset_scatter}. \n\n\n\n\\begin{figure*}[t!]\n\\centering\n\\includegraphics[width=0.95\\textwidth]{fig\/branchedmodel_cropped.pdf}\n\\caption{Branched model design. A central crop of the architecture diagram is shown for clarity (the full diagram is available within the documentation repo). The figure continues to the left and right in a similar pattern for all 20 pulse parameters ($\\alpha$).}\n\\label{fig:branchedmodel}\n\\end{figure*}\n\n\n\n\n\n\\subsection{Implementations with lookup tables}\\label{sec:lut}\n\nWhile it is significant that a machine-learning model can achieve comparable accuracy to a specific dataset, a designer may wonder why we cannot simply store the training dataset in a lookup table and use as an index the $\\beta$ value so we can obtain the optimized $\\alpha$ values when needed. Admittedly, such an implementation would take up a smaller area and have lower latency for this particular gate.\n\nHowever, the desired parameter domain may be much larger for more complicated gates. For example, in this initial set of experiments, we have only one gate parameter $\\beta{}$, which ranges from $-\\pi$ to $\\pi$. However, consider another gate that performs a rotation to \\textit{any} point on the Bloch sphere. The expected number of entries in the lookup table will increase exponentially as a function of the number of input parameters if the same per-parameter granularity is maintained. With an implementation based on neural networks, however, as we are still outputting only 20 pulse parameters, we may expect the network resource requirements to grow slower. For example, one could even imagine that we duplicate the network to produce 20 pulse parameters independently for each input parameter. Then, the designer can use another network to combine these pulse parameters into one final set of pulse parameters. Such a network architecture grows linearly in the number of gate parameters.\nFurthermore, these networks allow for interpolation and perhaps extrapolation. While a lookup table is constrained to the values stored, the ``intelligence'' of a neural network allows it to reasonably interpolate the pulse parameters for input gate parameters not seen during training.\n\nCurrent quantum computing systems implemented using lookup tables will round gate parameters to the closest matching entry in the table \\cite{liang20221}. The result is the application of lower-fidelity gates for the desired transformations, as the gate employed will have been optimized for different gate parameters than those of the desired gate. As a result, the accuracy of algorithmic results will decrease. For example, with the quantum approximate optimization algorithm (QAOA), we would expect the optimization results to be of lower quality, as gates are poorly approximated at each step~\\cite{qaoa}.\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.95\\columnwidth]{fig\/model-acc-resultgraph-testset_scatter.pdf}\n\\caption{Predicted-golden gate fidelity of large model. Test set points are marked by red circles, showing the ability of the model to interpolate. Fidelity exceeds $0.999$ for all test set $\\beta{}$.}\n\\label{fig:fidelity_largemodel_testset_scatter}\n\\end{figure}\n\n\n\n\\section{Conclusions}\n\nTo summarize the main contributions of this work, we have developed a neural network capable of predicting pulse parameters for a quantum gate (specifically, a rotation on the Bloch sphere around the x-axis) to the fidelity of $0.99$ and shown an FPGA implementation of this network that fits a small off-the-shelf development board. This implementation exhibits a latency of 175 ns and is pipelined with an initiation interval of 1 and a clock period of 5 ns. Our numerical and experimental setup is publicly available at \\cite{ml4quantum}.\n\n\n\nPreliminary results have shown that defining a machine learning model with a fidelity higher than $0.999$ is possible. We leave the FPGA deployment of such a model to future investigation together with the training with further-optimized training data, the augmentation of the dataset with data generated using additional random seeds, the design of a custom eFPGA-based system, the performance evaluation in extreme low-temperature environments, the comparison with lookup-table-based solutions, and the implementation of more complicated gates.\n\n\nOne may imagine that such models could be extended to control multi-qubit systems performing complex tasks. For example, application-specific hardware accelerators could be designed to capture multi-stage domain-specific operations in gates generated on-the-fly. Such abilities would allow for cheaper, more precise, and faster quantum computing systems that will be used to solve the problems of tomorrow. By moving the computation into the colder domains of the quantum computing system, the number of wires leading to the warmer domains can be reduced significantly (as we no longer need wires to transmit $\\alpha$ values calculated externally). Such wires are costly in the amount of energy they dissipate within the cold domains which are managed by machines of limited cooling capacity~\\cite{xue2021cmos}.\n\n\n\\subsection{Acknowledgements}\n\n\nWe thank Matt Otten for discussions on pulse parameters and neural networks. This work was conducted as part of the Spring 2022 seminar course ``CSEE E6868 - Embedded Scalable Platforms'' at Columbia University. DX thanks the TA Joseph Zuckerman for his help throughout the semester. AB\\\"O and GNP were partially supported by the DOE\/HEP QuantISED program grant ``HEP Machine Learning and Optimization Go Quantum,'' identification number 0000240323 and the DOE\/HEP QuantISED program grant ``QCCFP-QMLQCF Consortium,'' identification number DE-SC0019219. This manuscript has been authored by Fermi Research Alliance, LLC under Contract No. DE-AC02-07CH11359 with the U.S. Department of Energy, Office of Science, Office of High Energy Physics. We acknowledge the Fast Machine Learning collective as an open community of multi-domain experts and collaborators. This community was important for the development of this project.\n\n\n\n\n\n\n\n\\bibliographystyle{unsrtnat}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Acknowledgements}\n\n\\begin{acknowledgements}\nGT, CL\nand SR are grateful for support from the\nJulia Jacobi Foundation, the Swiss National\nScience Foundation (grant number\n200021\\_140472), and the European Research\nCouncil (grant number\n616305). DMW and MC acknowledge\nfunding from the Swiss National Science\nFoundation\n(Project ID 200021-159896). We are also\ngrateful for the generous allocation of CPU\ntime by CSCS under Project\nID s619.\n\\end{acknowledgements}\n\n\\section{Supporting Information}\n    The Supporting Information contains\n    further details of the following\n    aspects: the geometrical setup of\n    the SHS experiments; the computational details of the MD simulations; a discussion on the convergence of\n\tthe SHS simulations at low scattering wavevectors; a discussion of the effect\n\tof finite-size systems and of different water force-field\n\ton the computation of the intensity ratios.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn this proceedings we present an update of our investigations into a simple four-dimensional lattice theory comprising four massless reduced staggered fermions coupled through an SU(4)-invariant four-fermion interaction~\\cite{Catterall:2016dzf}.\nSystems of this sort have received considerable interest in recent years within the condensed matter community~\\cite{Fidkowski:2009dba, Morimoto:2015lua}, in the context of constructing models in which carefully chosen quartic interactions allow fermions to be gapped without breaking symmetries.\nThis feature of lattice four-fermion and Higgs--Yukawa models is predicted by strong-coupling arguments and was seen in earlier numerical calculations (including Refs.~\\cite{Stephenson:1988td, Hasenfratz:1988vc, Lee:1989xq, Lee:1989mi, Bock:1990cx, Hasenfratz:1991it, Golterman:1992yha} and references therein).\nThese works generically found a three-phase structure, with a symmetric massless `paramagnetic weak-coupling' (PMW) phase separated from a symmetric massive `paramagnetic strong-coupling' (PMS) phase by a wide intermediate `ferromagnetic' (FM) phase characterized by a symmetry-breaking bilinear fermion condensate.\nFirst-order transitions between these three phases left the symmetric massive PMS phase disconnected from the continuum limit.\n\nRecently, investigations of the three-dimensional version of the system we consider observed different behavior~\\cite{Ayyar:2014eua, Ayyar:2015lrd, Catterall:2015zua, He:2016sbs}.\nThree different numerical algorithms were used by these studies: fermion bags, rational hybrid Monte Carlo (RHMC) and quantum Monte Carlo.\nInstead of the three phases described above, these works identified a direct transition between the massless analog of the PMW phase and a massive PMS-like phase at strong coupling.\nThese two phases appear to be separated by a continuous phase transition with non-Heisenberg exponents, raising the possibility of a new continuum limit at strong coupling.\n\nSince the system we study possesses different exact lattice symmetries than those considered earlier, \\textit{a priori} it is possible that this two-phase structure may persist in four dimensions.\nThe first work exploring the four-dimensional theory reported the reappearance of a broken FM phase~\\cite{Ayyar:2016lxq, Ayyar:2016nqh}.\nHowever, in contrast to the earlier studies this intermediate phase was very narrow, and the transitions bounding it appeared consistent with second-order criticality with mean-field exponents.\nThese conclusions were not based on explicit measurements of bilinear condensates, but rather inferred from the volume scaling of a certain susceptibility.\nOur own work in \\refcite{Catterall:2016dzf} added source terms to the action in order to more directly address whether spontaneous symmetry breaking associated with the formation of specific bilinear condensates takes place.\nWe observed long-range correlations in the narrow critical region between the PMW and PMS phases, but did not observe spontaneous symmetry breaking and could not resolve whether this critical region corresponded to an intermediate phase or a single broad transition.\n\nSince publishing \\refcite{Catterall:2016dzf} we have discovered and corrected two factor-of-two errors in our publicly available code,\\footnote{\\texttt{\\href{https:\/\/github.com\/daschaich\/fourfermion}{github.com\/daschaich\/fourfermion}}} both related to the reality of our pseudofermions.\nBriefly, the normalization of the gaussian pseudofermion fields $\\Phi$ generated at the start of each RHMC trajectory was $\\sqrt{2}$ times too large, while the fermion action itself should have involved $\\Phi^T \\left(\\ensuremath{M^{\\dag}}  M\\right)^{-1 \/ 2} \\Phi$ rather than $\\Phi^T \\left(\\ensuremath{M^{\\dag}}  M\\right)^{-1 \/ 4} \\Phi$, where $M$ is the fermion operator.\nThese two errors largely counteracted each other, which allowed them to slip past our software tests.\\footnote{We thank Jarno Rantaharju for independently checking our results, which helped us track down these problems.}\nFollowing a short review of the lattice theory in \\secref{sec:review}, in \\secref{sec:results} we present preliminary data obtained with the corrected code, finding broad consistency with our published results.\nThe main change is a developing signal for spontaneous symmetry breaking in the critical region, potentially consistent with the results of Refs.~\\cite{Ayyar:2016lxq, Ayyar:2016nqh}.\nWe conclude in \\secref{sec:conc} with our plans to solidify this result and more carefully study the two transitions that a broken intermediate phase would imply.\n\n\n\n\\section{\\label{sec:review}Review of the lattice theory}\nAs described above, we consider four massless reduced staggered fermions in four dimensions.\nThe lattice action (with sums over repeated indices)\n\\begin{equation}\n  \\label{eq:action}\n  S = \\sum_x \\left[\\frac{1}{2} \\psi^a(x) \\eta_{\\mu}(x) \\ensuremath{\\Delta} _{\\mu}^{ab} \\psi^b(x) - \\frac{1}{4} G^2 \\ensuremath{\\epsilon} _{abcd} \\psi^a(x) \\psi^b(x) \\psi^c(x) \\psi^d(x)\\right]\n\\end{equation}\ncontains a single-site SU(4)-invariant four-fermion term.\nHere $\\eta_{\\mu}(x) = \\left(-1\\right)^{\\sum_{i < \\mu} x_i}$ is the usual staggered fermion phase while $\\ensuremath{\\Delta} _{\\mu}^{ab} \\psi^b(x) = \\frac{1}{2} \\ensuremath{\\delta} ^{ab} \\Big\\{\\psi^b(x + \\ensuremath{\\hat\\mu} ) - \\psi^b(x - \\ensuremath{\\hat\\mu} )\\Big\\}$.\nThe action possesses exact global symmetries under the transformations\n\\begin{align}\n  \\psi(x) & \\to e^{i\\ensuremath{\\epsilon} (x) \\ensuremath{\\alpha} } \\psi(x)      & & \\mbox{with} & \\ensuremath{\\epsilon} (x)      & = \\left(-1\\right)^{\\sum_i x_i} \\quad \\mbox{and} \\quad \\ensuremath{\\alpha}  \\in \\mathrm{su(4)} \\cr\n  \\psi(x) & \\to \\ensuremath{\\Gamma}  \\psi(x)                   & & \\mbox{with} & \\ensuremath{\\Gamma}           & \\in \\Big\\{1, -1, i\\ensuremath{\\epsilon} (x), -i\\ensuremath{\\epsilon} (x)\\Big\\}                                    \\\\\n  \\psi(x) & \\to \\xi_{\\mu}(x) \\psi(x + \\ensuremath{\\hat\\mu} ) & & \\mbox{with} & \\xi_{\\mu}(x) & = \\left(-1\\right)^{\\sum_{i > \\mu} x_i}.                                      \\nonumber\n\\end{align}\nThe first of these corresponds to the above-mentioned SU(4) symmetry.\nThe second is a $Z_4$ subgroup of the usual U(1) symmetry, which is all that the four-fermion interaction preserves.\nFinally the staggered shift symmetries in the third line can be considered a discrete remnant of continuum chiral symmetry~\\cite{Bock:1992yr}.\n\nSince reduced staggered fermions involve no independent \\ensuremath{\\overline\\psi}  fields, these symmetries strongly constrain the possible bilinear terms that can arise in the lattice effective action.\nSingle-site bilinears of the form $\\psi^a(x)\\psi^b(x)$ violate the SU(4) and $Z_4$ symmetries, while the SU(4)-invariant multilink bilinear operators\n\\begin{align}\n  \\label{eq:ops}\n  O_1 & = \\sum_{x,\\,\\mu} m_{\\mu} \\ensuremath{\\epsilon} (x) \\xi_{\\mu}(x) \\psi^a(x) S_{\\mu} \\psi^a(x)                                   &\n  O_3 & = \\sum_{x,\\,\\mu,\\,\\nu,\\,\\ensuremath{\\lambda} } m_{\\mu\\nu\\ensuremath{\\lambda} } \\xi_{\\mu\\nu\\ensuremath{\\lambda} }(x) \\psi^a(x) S_{\\mu} S_{\\nu} S_{\\ensuremath{\\lambda} } \\psi^a(x)\n\\end{align}\nviolate the shift symmetries~\\cite{vandenDoel:1983mf, Golterman:1984cy}.\nIn these expressions\n\\begin{align*}\n  \\label{eq:sym}\n  \\xi_{\\mu\\nu\\ensuremath{\\lambda} }(x) & \\equiv \\xi_{\\mu}(x) \\xi_{\\nu}(x + \\ensuremath{\\hat\\mu} ) \\xi_{\\ensuremath{\\lambda} }(x + \\ensuremath{\\hat\\mu}  + \\ensuremath{\\hat\\nu} ) &\n  S_{\\mu} \\psi(x)    & = \\psi(x + \\ensuremath{\\hat\\mu} ) + \\psi(x - \\ensuremath{\\hat\\mu} )\n\\end{align*}\nand $m_{\\mu\\nu\\ensuremath{\\lambda} }$ is totally antisymmetric in its indices.\nIn the absence of explicit symmetry-breaking external sources, bilinear condensates can only form if the corresponding lattice symmetries break spontaneously.\n\nEven in the absence of symmetry-breaking bilinear condensates, a strong-coupling expansion predicts non-zero masses for both fermionic and bosonic excitations.\nThe leading term in this expansion corresponds to the static $G \\to \\infty$ limit in which the kinetic operator is dropped.\nIn the partition function we expand the exponential of the four-fermion term in powers of $G$, obtaining\n\\begin{equation}\n    Z \\sim \\left[6G^2 \\int d\\psi^1(x)d\\psi^2(x) d\\psi^3(x) d\\psi^4(x) \\psi^1(x)\\psi^2(x)\\psi^3(x)\\psi^4(x)\\right]^V\n\\end{equation}\nfor lattice volume $V$, which corresponds to saturation by a single-site four-fermion condensate.\n\nA straightforward computation of the fermion propagator $F(x) = \\vev{\\psi^1(x) \\psi^1(0)}$, following the procedure described in \\refcite{Eichten:1985ft}, produces the momentum-space expression\n\\begin{align}\n  F(p) & = \\frac{i\\sqrt{6G^2} \\sum_{\\mu} \\sin p_{\\mu}}{\\sum_{\\mu}\\sin^2 p_{\\mu} + m_F^2} & & \\mbox{with} & m_F^2 & = 4 \\left(6G^2\\right)^2 - 2.\n\\end{align}\nAn analogous calculation for the bosonic propagator $B(x) = \\vev{b(x) b(0)}$ of the single-site fermion bilinear $b \\equiv \\psi^1\\psi^2 + \\psi^3\\psi^4$ leads to\n\\begin{align}\n  B(p) & = \\frac{8 \\left(6G^2\\right)}{4 \\sum_{\\mu}\\sin^2 (p_{\\mu} \/ 2) + m_B^2} & & \\mbox{with} & m_B^2 & = 4 \\left(6G^2\\right) - 8.\n\\end{align}\nIn addition to predicting massive fermionic and bosonic excitations at strong coupling, this analysis suggests an interpretation for the symmetric mass generation.\nNamely, this may correspond to the condensation of a bilinear formed from the original elementary fermion $\\psi^a$ and the composite fermion $\\Psi^a \\equiv \\ensuremath{\\epsilon} _{abcd} \\psi^b \\psi^c \\psi^d$ that transforms in the complex conjugate representation of the SU(4) symmetry.\nThe formation of such a four-fermion condensate is clearly a non-perturbative phenomenon invisible in weak-coupling perturbation theory, which motivates our numerical lattice investigations.\n\nAs usual, our RHMC calculations use an auxiliary real scalar field $\\ensuremath{\\sigma} _+^{ab} = -\\ensuremath{\\sigma} _+^{ba}$ to bring the action~(\\ref{eq:action}) into a form quadratic in the fermion fields,\n\\begin{equation*}\n  S = \\sum_x \\left[\\psi^a(x) \\left(\\eta.\\ensuremath{\\Delta} ^{ab} + G\\ensuremath{\\sigma} _+^{ab}(x)\\right) \\psi^b(x) + \\frac{1}{4}\\left(\\ensuremath{\\sigma} _+^{ab}(x)\\right)^2\\right] = \\sum_x \\left[\\psi^a(x) M^{ab}(x) \\psi^b(x) + \\frac{1}{4}\\left(\\ensuremath{\\sigma} _+^{ab}(x)\\right)^2\\right],\n\\end{equation*}\ndefining the fermion operator $M^{ab}(x) \\equiv \\eta.\\ensuremath{\\Delta} ^{ab} + G\\ensuremath{\\sigma} _+^{ab}(x)$.\nThe subscript indicates that $\\ensuremath{\\sigma} _+^{ab}$ is self-dual,\n\\begin{equation}\n  \\label{eq:proj}\n  \\ensuremath{\\sigma} _+^{ab} = \\frac{1}{2}\\left(\\ensuremath{\\sigma} ^{ab} + \\frac{1}{2} \\ensuremath{\\epsilon} _{abcd} \\ensuremath{\\sigma} ^{cd}\\right) \\equiv P_+^{abcd} \\ensuremath{\\sigma} ^{cd}.\n\\end{equation}\nNow we can integrate over the fermions to produce the pfaffian $\\ensuremath{\\text{pf}\\,} {M(\\ensuremath{\\sigma} _+)}$, which turns out to be positive semi-definite.\nThis follows from the fact that $M$ is real, anti-symmetric, and invariant under one of the SU(2) subgroups of the $\\mathrm{SO(4)} \\simeq \\mathrm{SU(2)} \\times \\mathrm{SU(2)}$ global symmetry of the quadratic action above.\nThe eigenvalues of $M$ are therefore pure imaginary and come in pairs $i\\ensuremath{\\lambda} $ and $-i\\ensuremath{\\lambda} $, both of which are doubly degenerate due to the invariance under this SU(2).\nWe have checked this conclusion numerically.\nIt forbids sign changes in the pfaffian, which would correspond to an odd number of eigenvalues passing through zero as $\\ensuremath{\\sigma} _+$ varies.\n\n\n\n\\section{\\label{sec:results}Numerical results for the phase diagram}\nIn order to directly search for spontaneous symmetry breaking we have augmented the lattice action~(\\ref{eq:action}) by adding three source terms,\n\\begin{equation}\n  \\label{eq:sources}\n  \\ensuremath{\\Delta}  S = \\sum_{x,\\,a,\\,b} \\Big(m_1 + \\ensuremath{\\epsilon} (x)m_2\\Big) \\left[\\psi^a(x) \\psi^b(x)\\right]_+ \\ensuremath{\\Sigma} ^{ab} + m_3\\sum_{x,\\,\\mu,\\,a} \\ensuremath{\\epsilon} (x) \\xi_{\\mu}(x) \\psi^a(x) S_{\\mu} \\psi^a(x).\n\\end{equation}\nAs in \\eq{eq:proj} we consider the self-dual part of the single-site bilinear in the first set of terms, which we couple to the SU(4)-breaking source $\\displaystyle \\ensuremath{\\Sigma} ^{ab} = \\left(\\begin{array}{cc} i\\ensuremath{\\sigma} _2 & 0 \\\\ 0 & i\\ensuremath{\\sigma} _2\\end{array}\\right)$.\nIn this proceedings we work with $m_1 = 0$, considering only the ``staggered'' single-site bilinear with coupling $m_2$.\nThe latter operator breaks all the exact symmetries of the action but appears as a rather natural mass term when the model is rewritten in terms of two full staggered fields.\nThe final term corresponds to the shift-symmetry-breaking one-link operator $O_1$ in \\eq{eq:ops}.\n\n\\begin{figure}[btp]\n  \\includegraphics[width=0.48\\linewidth]{plus} \\hfill \\includegraphics[width=0.48\\linewidth]{sus}\n  \\caption{\\label{fig:transition}\\textbf{Left:} $\\frac{1}{4} \\vev{\\ensuremath{\\sigma} _+^2} - \\frac{3}{2}$ serves as a proxy for the four-fermion condensate, increasing from zero at weak four-fermion coupling $G$ to unity at strong coupling.  \\textbf{Right:} The staggered susceptibility $\\chi$ peaks in a narrow critical region $1 \\ensuremath{\\lesssim}  G \\ensuremath{\\lesssim}  1.1$.  Both are measured on $4^4$ lattices with $m_2 = 0.01$ while $m_1 = m_3 = 0$.}\n\\end{figure}\n\nAs mentioned in the introduction, our corrected numerical results are broadly consistent with those published in \\refcite{Catterall:2016dzf}.\n\\fig{fig:transition}, for example, reproduces the behavior shown in Figs.~1 and 14 of \\refcite{Catterall:2016dzf} for two observables that are sensitive to the transition from weakly coupled free fields to strongly coupled four-fermion condensates.\nThe square of the auxiliary field $\\frac{1}{4} \\ensuremath{\\sigma} _+^2 = \\frac{1}{2} \\sum_{a < b}\\left(\\ensuremath{\\sigma} _+^{ab}\\right)^2$ in the left plot serves as a proxy for the four-fermion condensate.\nA simple analytic calculation predicts $\\frac{1}{4} \\vev{\\ensuremath{\\sigma} _+^2} \\to \\frac{3}{2}$ as $G \\to 0$ and $\\frac{1}{4} \\vev{\\ensuremath{\\sigma} _+^2} \\to \\frac{5}{2}$ as $G \\to \\infty$, which our numerical results reproduce.\nWe see a continuous interpolation between these two limits, but this is not significant given the small $4^4$ lattice volume and non-zero $m_2 = 0.01$ considered so far.\n\nThe right plot of \\fig{fig:transition} shows the staggered susceptibility\n\\begin{align}\n  \\chi & = \\frac{1}{V}\\left(\\vev{O_{\\text{stag}}^2} - \\vev{O_{\\text{stag}}}^2\\right) &\n  O_{\\text{stag}} & = \\sum_x \\ensuremath{\\epsilon} (x) \\left[\\psi^0(x) \\psi^1(x)\\right]_+\n\\end{align}\ncomputed on these same $4^4$ ensembles.\nAs in Fig.~14 of \\refcite{Catterall:2016dzf} we see a clear peak in the susceptibility centered around $G_c \\approx 1.05$.\nThe height of this peak is significantly larger than we saw before, but this is related to the non-zero $m_2 = 0.01$ we use in \\fig{fig:transition}.\nBoth plots in this figure indicate a narrow critical region in the same range $1 \\ensuremath{\\lesssim}  G \\ensuremath{\\lesssim}  1.1$ that we observed previously.\n\n\\begin{figure}[btp]\n  \\centering\n  \\includegraphics[width=0.5\\linewidth]{eig_m001_m0}\n  \\caption{\\label{fig:eig}The smallest (squared) eigenvalue of the fermion operator vs.\\ $1 \/ L$ on log--log axes, for fixed $G = 1.05$ and $m_2 = 0.001$ with $m_1 = m_3 = 0$.  A power-law fit to the three largest volumes produces $\\ensuremath{\\lambda} _0^2 \\propto L^{-9 \/ 2}$.}\n\\end{figure}\n\nWe are currently scanning in $G$ for larger volumes with all three $m_i = 0$, which we expect will reproduce the volume scaling of the peak susceptibility ($\\chi_{\\text{peak}} \\sim L^4$) reported by \\refcite{Ayyar:2016lxq}.\nIn \\refcite{Catterall:2016dzf} we identified two other signals of long-range correlations in this critical region: the mass of a composite boson was very small throughout $1 \\leq G \\leq 1.1$ while the smallest eigenvalue of the fermion operator decreased rapidly with the volume in this regime.\nWhile we have not yet repeated our bosonic two-point function analyses, \\fig{fig:eig} shows that the smallest eigenvalue at $G = 1.05$ continues to decrease rapidly, $\\ensuremath{\\lambda} _0 \\propto L^{-9 \/ 4}$.\nWe use a small but still non-zero $m_2 = 0.001$, which significantly increases $\\ensuremath{\\lambda} _0$ compared to the $m_2 = 0$ results shown in Fig.~5 of \\refcite{Catterall:2016dzf}.\nThe corresponding reduction in critical slowing down has allowed us to investigate larger $L = 16$ than we could easily reach with $m_2 = 0$.\n\nFinally we revisit our search for spontaneous symmetry breaking in the critical region that may correspond to a narrow intermediate phase.\nFixing $G = 1.05$ (and $m_1 = 0$), we compute the vacuum expectation values of the staggered site and one-link bilinears as functions of $m_2$, $m_3$ and the lattice volume.\nThe presence of the source terms leads to non-zero vevs for the corresponding operators, which on a fixed lattice volume must vanish as the couplings are sent to zero, due to the exact lattice symmetries that appear in that limit.\nA signal of spontaneous symmetry breaking would be a condensate that grows with the volume for small values of the external source, which would allow for the possibility that the condensate remains finite in the thermodynamic limit as the source is removed.\n\n\\begin{figure}[btp]\n  \\includegraphics[width=0.48\\linewidth]{bi_nolink} \\hfill \\includegraphics[width=0.48\\linewidth]{bi_link}\n  \\caption{\\label{fig:bi}The staggered site bilinear vs.\\ $m_2$ for $G = 1.05$ and $m_1 = 0$.  In the left plot we consider $L = 4$, 8, 12 and 16 with no source for the one-link bilinear, $m_3 = 0$.  In the right plot we set $m_3 = m_2$ and study $L = 4$, 8 and 12 as in \\protect\\refcite{Catterall:2016dzf}.}\n\\end{figure}\n\nIn \\fig{fig:bi} we see initial signs of such behavior in the staggered site bilinear.\nThe two plots in the figure differ only in the value of the one-link coupling: $m_3 = 0$ in the left plot while the right plot has $m_3 = m_2$ as in \\refcite{Catterall:2016dzf}.\nThe staggered site bilinear shows no visible dependence on $m_3$, with results for $4 \\leq L \\leq 16$ following a single curve for large $m_2 \\ensuremath{\\gtrsim}  0.04$.\nIn both plots we can see the smallest-volume $4^4$ results departing from this curve and moving towards zero for $m_2 \\ensuremath{\\lesssim}  0.03$, while the next $8^4$ results follow suit around $m_2 \\ensuremath{\\lesssim}  0.003$.\nHowever, for $m_3 = 0$ in the left plot we have begun investigating larger $L = 16$, and find that the corresponding staggered site bilinear is no larger than the $L = 12$ results for all $m_2 \\geq 0.001$ we have reached so far.\nWhile the small-volume behavior in \\fig{fig:bi} supports spontaneous SU(4) symmetry breaking at $G = 1.05$, we hope to strengthen this conclusion with more data on larger volumes in the near future.\n\n\\begin{figure}[btp]\n  \\centering\n  \\includegraphics[width=0.5\\linewidth]{link}\n  \\caption{\\label{fig:link}The one-link bilinear vs.\\ $m_3 = m_2$ for $L = 4$, 8 and 12 at $G = 1.05$ and $m_1 = 0$.}\n\\end{figure}\n\n\\fig{fig:link} carries out the same exercise for the one-link bilinear, using the same $m_3 = m_2$ ensembles considered in the right plot of \\fig{fig:bi}.\n(When $m_3 = 0$ the one-link condensate vanishes.)\nHere we do not see any sign that the staggered shift symmetries break spontaneously.\nThe results are qualitatively the same as those in Fig.~12 of \\refcite{Catterall:2016dzf}, with no significant volume dependence visible for any $m_3 \\leq 0.1$.\n\n\n\n\\section{\\label{sec:conc}Conclusions and next steps}\nWhile the system we study is perhaps the simplest four-fermion theory in four dimensions, it exhibits interesting behavior that motivates further work, both theoretical and computational.\nWith our corrected code we observed the same large-scale phase structure as in \\refcite{Catterall:2016dzf}, with a narrow critical region around $1 \\ensuremath{\\lesssim}  G \\ensuremath{\\lesssim}  1.1$ separating the symmetric massless weak-coupling (PMW) phase from the symmetric massive strong-coupling (PMS) phase.\nWe continue to see signs of long-range correlations in this critical region, and have now begun to see a developing signal consistent with spontaneous SU(4) symmetry breaking at $G = 1.05$.\n\nWe are already working to improve \\fig{fig:transition} by repeating our scans in the four-fermion coupling $G$ on larger volumes with zero external sources.\nThis will provide another look at the finite-size scaling of the peak in the staggered susceptibility, and hopefully will allow us to resolve the two transitions that a spontaneously broken intermediate phase would imply.\nFurther studies would then be needed to distinguish whether the transitions are continuous or first order, and to estimate critical exponents in the former case, with the goal of exploring whether they may provide the possibility of a new continuum limit for strongly interacting fermions.\n\nIt is also important to improve the scans in $m_2$ shown in \\fig{fig:bi}, to strengthen the developing signs of spontaneous symmetry breaking at $G = 1.05$.\nRepeating this exercise in the PMW and PMS phases (e.g., for $G \\approx 0.85$ and 1.25, respectively) should provide useful contrasts that may clarify the nature of the critical regime.\nIt seems likely we will need to consider smaller $m_2 \\ll 10^{-3}$ at $G = 1.05$, which will be challenging due to the critical slowing down we observe in the transition region when $m_2 = 0$.\nThe rapid decrease in the smallest eigenvalues of the fermion operator corresponds to an increasing condition number in each multi-mass conjugate gradient inversion of $\\ensuremath{M^{\\dag}}  M$, which dominates the computational cost of the RHMC algorithm.\nIt may be necessary to implement significant extensions to the code to make these calculations practical, for example applying staggered deflation or multigrid techniques.\n\nIndependently of these numerical investigations, there is also work ongoing to explore the conceptual issues connected to the observed behavior.\nFor example, it has been proposed~\\cite{You:2014vea, BenTov:2015gra, Wang:2013yta} that similar quartic interactions can be used in the context of domain wall fermions to realize a lattice regularization of chiral gauge theories along the lines originally proposed by \\refcite{Eichten:1985ft}.\nAlthough these proposals describe non-relativistic fermions using hamiltonian language, it is nevertheless intriguing that the sixteen Majorana fermions they require match the sixteen Majorana fermions that are expected at weak coupling in this lattice theory.\nHowever, it is not clear to the authors whether these proposals circumvent all the difficulties described in \\refcite{Poppitz:2010at}.\nEven the possibility of a new strongly interacting continuum limit is subtle.\nThe reduced staggered fermions used to define the lattice theory transform under a twisted group comprising both Lorentz and flavor symmetries~\\cite{Banks:1982iq}, suggesting that even if a new fixed point exists it might not be Lorentz invariant.\n\nMore recently there have been interesting proposals concerning the novel phase structures of the three- and four-dimensional theories.\nIn three dimensions, for example, the apparent direct transition between the weakly and strongly coupled symmetric phases might be understood within the theoretical framework of deconfined quantum criticality~\\cite{You:2017ltx}.\nIn four dimensions \\refcite{Catterall:2017ogi} argued that the fermion mass in the PMS phase may result from the proliferation of topologically non-trivial defects with non-zero Hopf invariant, following a similar logic to Witten's treatment of the two-dimensional Thirring model~\\cite{Witten:1978qu}.\nWe look forward to further work on this subject, in particular continuing interplay between non-perturbative numerical investigations and conceptual considerations.\n\n\n\n\\vspace{10 pt}\n\\noindent {\\sc Acknowledgments:}~We thank Shailesh Chandrasekharan and Jarno Rantaharju for useful discussions.\nThis work is supported in part by the U.S.~Department of Energy, Office of Science (DOE), Office of High Energy Physics, under Award Number DE-SC0009998.\nNumerical calculations were carried out on the DOE-funded USQCD facilities at JLab.\n\n\n\n\\raggedright\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}