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+{"text":"\\section{Introduction}\n\nThe past decade has seen major developments in the field of machine learning, and societal applications in  health-care, autonomous vehicles, and language processing are becoming commonplace \\cite{jordan2015machine}. The impact of machine learning on basic research has been just as significant, and the use of advanced algorithmic tools in data analysis has resulted in new insights into many areas of science. In physics, there has been particular interest applying the tools of machine learning to study dynamical complex systems which evolve in time. These systems exhibit extreme sensitivity to small variations of the governing parameters, and the use of conventional numerical methods to understand and potentially control these dynamics is challenging.  \n\nNonlinear pulse propagation in optical fibre waveguides is known to exhibit highly complex evolution, and machine learning methods have been applied in a variety of ways to both optimize and analyze their spectrum or temporal intensity profile at the fibre output. For example from a feedback and control perspective, evolutionary algorithms (which are typically slow to converge) have been used in experiments optimizing particular characteristics of supercontinuum sources \\cite{wetzel2018customizing, michaeli2018genetic}, as well as the experimental control of mode-locked fibre lasers \\cite{andral2015fiber,pu2019intelligent,dudleymeng2020,kokhanovskiy2019machine2}.  Machine learning using neural networks has also been applied to classify experimentally different regimes of nonlinear propagation in modulation instability experiments \\cite{narhi2018machine} or to determine the duration of short pulses from a fibre laser \\cite{kokhanovskiy2019machine}. Applications to the control of mode-locking \\cite{baumeister2018,kokhanovskiy2019machine} and pulse shaping \\cite{finot2018nonlinear} have also been demonstrated numerically. Yet, all these applications have been restricted either to (slow) genetic algorithms or to feed-forward neural network architectures limited to determine the correspondence between a given input and some single output parameter. \n\nMore generally, experiments in optical fibres are of very wide interest in nonlinear science since they provide a convenient means of studying nonlinear dynamics common to many nonlinear Schr\\\"{o}dinger equation (NLSE) systems including hydrodynamics, plasmas, and Bose-Einstein condensates.  However, because propagation in an NLSE system depends sensitively on both the input pulse and fibre characteristics, the design and analysis of experiments require extensive numerical simulations based on the numerical integration of the NLSE or its extensions. This is computationally-demanding and creates a severe bottleneck in using numerical techniques to design or optimize experiments in real-time. \n\nIn this paper, we present a solution to this problem using machine-learning to predict complex nonlinear propagation in optical fibres with a recurrent neural network, bypassing the need for direct numerical solution of a governing propagation model.  The general context of our work is the recent development of machine learning approaches exploiting  \\emph{knowledge-based} and \\emph{model-free} methods to forecast and thus control complex evolving dynamics. Knowledge-based (or physics-informed) methods rely on some a priori knowledge of the mathematical model governing the physical system, and they perform especially well in capturing nonlinear dynamics \\cite{brunton2016discovering,raissi2018deep, raissi2019physics}. In contrast, model-free forecasting is a purely \\emph{data-driven} approach where a neural-network structure will learn the system dynamical behavior from a set of training data, without any prior knowledge of the physics of the system or any underlying governing equation(s). Model-free methods have been particularly successful in forecasting spatio-temporal dynamics of physical systems exhibiting high-dimensional chaos, instabilities and turbulence \\cite{vlachas2018data, vlachas2019forecasting, pandey2020reservoir}, as well as reproducing the propagation dynamics of certain analytical solutions of the NLSE \\cite{jiang2019model}.  \n\nOur objective here is to significantly extend the use of model-free methods in nonlinear physics by showing how a long short-term memory (LSTM) recurrent neural network can fully reproduce the complex nonlinear dynamics of ultrashort pulse evolution in optical fibre governed by an NLSE system.  We study two particular cases of practical importance: high power pulse compression associated with the generation of Peregrine-soliton structures, and broadband optical supercontinuum generation.  In the first case, we  show how the network accurately predicts the temporal and spectral evolution of higher-order solitons and the appearance of the Peregrine soliton from a transform-limited intensity profile, and we also show how the predicted results agree with reported experimental measurements \\cite{tikan2017universality}. We then expand our analysis to even more complex dynamics and show how the network can also predict the full development of an octave-spanning supercontinuum with fine details in the spectral and temporal domains. These results represent a significant extension of model-free methods applied to nonlinear optics, with potential important impact for high-field physics, nonlinear spectroscopy, and precision frequency comb metrology.  Moreover, we anticipate that our results will stimulate similar studies in all areas of physics where NLSE-like dynamics play a governing role.  \n\n\n\n\n\\section{Model-free modeling of nonlinear propagation dynamics}\n\nThe propagation of light in an optical fibre can be represented as a sequence of electric field complex amplitude distributions (spectral or temporal) at different points along the propagation path in the fibre. The amplitude at any specific propagation distance is naturally determined by the evolution which precedes it, and modelling this evolution is conventionally carried out by numerically integrating a governing NLSE model over a large number of elementary steps \\cite{agrawal}. Unfortunately, this conventional approach can be extremely time-consuming.\n\nHere, we show that such a direct numerical approach can in fact be replaced with model-free forecasting using a recurrent neural network (RNN). RNNs are a particular class of neural network that possess internal memory, allowing them to account for long-term dependencies and thus to robustly identify patterns in sequential data \\cite{Lipton2015RNN}. The fact that RNNs intrinsically allow modelling of dynamic behavior makes them particularly adapted to the processing and predictions of time-series with applications in speech recognition, predictive texting, handwriting recognition, natural language processing, or stock market analysis. And they are equally a natural choice to predict the evolution of nonlinear propagation dynamics as a high power optical field propagates in an optical fibre.   \n\nThe particular form of RNN we use is the long short-term memory (LSTM) cell architecture \\cite{hochreiter1997long}. Although other approaches such as reservoir computing or the gated recurrent unit would also be possible, our choice of LSTM is based on its simplicity of implementation and demonstrated success in various applications \\cite{Carleoreview2019,reviewRNN2019}.  We train the network to be able to separately and independently forecast the evolution of temporal and spectral intensity during nonlinear pulse propagation in optical fibre, based only on the initial condition of a transform-limited pulse. Of course physically, the temporal and spectral field characteristics are tightly coupled, and it is therefore remarkable that the network is able to learn independently the temporal and spectral evolution dynamics using only intensity data.  In order to teach the network the pulse propagation dynamics, initial training is performed using ensembles of temporal and spectral intensity evolution maps, generated numerically using simulations of the NLSE (or its generalized version the GNLSE) for a range of input pulse characteristics.  In order to reduce the computational load during training, the simulation profiles are downsampled along both the propagation direction, and the temporal and spectral dimensions (see Methods).  \n\n\n\\begin{figure}[!ht]\n  \\centering\n  \\includegraphics[width=\\linewidth]{schmatic13.pdf}\n\\caption{(a) Schematic of the recurrent neural network architecture used showing: the input layer, the long short-term memory (LSTM) recurrent layer, two hidden (dense) layers, and the output layer. (b) The neural network uses the spectral (or temporal) intensity profiles $\\mathbf{X}_{z-1}$ from the ten previous intensity profiles $h_{z-10\\Delta z}..h_{z-\\Delta z}$ in the evolution to yield the subsequent spectrum $h_{z}$. Each intensity profiles $h$ consists of $B$ intensity bins denoted as $x^{k}$ where $k$ indicates the bin number. (c) The LSTM cell receives the cell input, hidden and cell states from the previous step as an input, and output of the cell is the new hidden state that is also passed on to the next prediction step along with the new cell state. $x_i$ is the cell input where $i = z, z - \\Delta z$, $h_i$ denotes the hidden state and $c_i$ the cell state. The yellow rectangles denote layer operations and the orange circles denote pointwise operations. See Methods for more details on the number of nodes used per layer, activation functions etc. More details and definition of the different cell elements are given in the Methods.} \n\\label{fig:schmatic}\n\\end{figure}\n\nA general schematic of the RNN is shown in Fig. \\ref{fig:schmatic}(a) and an illustration of the training stage is shown in Fig. \\ref{fig:schmatic}(b). Ten consecutive temporal or spectral intensity profiles $h_{z-10\\Delta z}..h_{z-\\Delta z}$ (i.e. the evolution from distance $z-10\\Delta z$ to $z-\\Delta z$) are fed to the RNN. Here $\\Delta z$ corresponds to the sampling distance along the propagation direction (see Methods). The choice to feed the network with ten consecutive intensity profiles at propagation interval $\\Delta z$ was found to be a good heuristic compromise between speed and performance (see Methods). These intensity profiles are then passed to the LSTM layer consisting of cells (Fig. \\ref{fig:schmatic}(c)) governed by a specific algorithm (see Methods). Essentially, the LSTM layer uses 3 different types of information to predict the (spectral or temporal) intensity profile $h_{z}$ at distance $z$: (i) the intensity profile $h_{z-\\Delta z}$ at distance $z-\\Delta z$ which is the input of the LSTM layer, the hidden state of the layer corresponding to the predicted intensity profile $h_{z-2\\Delta z}$ at distance $ z-2\\Delta z$, and the cell state which contains the long-term dependency information from the intensity profiles $h_{z-10\\Delta z}..h_{z-3\\Delta z}$ corresponding to the evolution from distance $z-10\\Delta z$ to $z-3\\Delta z$.\n\nThe output of the LSTM layer is subsequently fed to a fully connected feed-forward neural network with two hidden (dense) layers whose function is to further improve the predicted intensity at distance $z$. The prediction made by the RNN (output layer) is compared with the intensity profile from the NLSE (or it generalized version GNLSE) simulations. The error is backpropagated to the weights and biases of the network nodes (both dense and LSTM layers) that are subsequently adjusted to minimize the prediction error. The RNN cycle is then initiated again with an updated input consisting of the consecutive temporal or spectral intensity profiles $h_{z-9\\Delta z}..h_{z}$ till the full evolution is predicted. Note that the RNN loop is initiated with a ``cold start'' where the input sequence contains only the spectral or temporal intensity profile of pulses injected into the fibre (replicated ten times). In the prediction phase, the RNN model is tested using a separate set of temporal and spectral evolution data that was not used in the training phase.\n\n\\section{Results}\n\\label{sec:results}\n\\subsection{Higher-order soliton compression}\nWe begin by training the RNN to model the propagation of picosecond pulses in the anomalous dispersion regime of a highly nonlinear fibre. This propagation regime is of particular significance as it is associated with extreme self-focusing dynamics and practical ``higher-order soliton'' pulse compression schemes \\cite{agrawal}. Moreover, the dynamics of this nonlinear temporal compression have been recently shown to be associated with the emergence of the celebrated Peregrine soliton that appears in the semiclassical limit of the NLSE \\cite{tikan2017universality}.  \n\nThe training data was generated by performing 3,000 NLSE numerical simulations of propagation in 13~m of fibre using initial conditions of transform-limited hyperbolic-secant input pulses. The fibre parameters were kept constant between simulations and corresponded to experiments performed around 1550~nm \\cite{tikan2017universality}.  On the other hand, we varied the pulse duration $\\Delta\\,\\tau$ (FWHM) and peak power $P_0$  uniformly over the ranges 0.77--1.43~ps and 18.6--34.2 W, respectively.  This yields a variation in soliton number from $N=3.5-8.9$ where $N^2 = \\gamma P_0 T_0^2\/\\beta_2$ with $\\gamma$, $\\beta_2$ the fibre nonlinear and group velocity dispersion parameters respectively, and $T_0 = \\Delta\\,\\tau  \/1.763 $.  See Methods for further details.   \n\n\\begin{figure}[!ht]\n  \\centering\n  \\includegraphics[width=\\linewidth]{PS_NLSE_v4.pdf}\n \n\\caption{Temporal intensity evolution of a 1.1~ps (full width at half maximum) pulse with 26.3~W peak power corresponding to an $N=6$ soliton injected into the anomalous dispersion regime of a 13~m long highly nonlinear fibre. The panels shows the result of NLSE numerical simulation (left), RNN prediction (middle), and relative difference (right). The RNN predictions use only the injected pulse intensity profile as input.}\n\\label{fig:pstime}\n\\end{figure}\n\nWe first illustrate the results obtained when training the network to model the temporal intensity evolution. Figure~\\ref{fig:pstime} compares the evolution of the temporal intensity simulated using the NLSE (left panel) with that predicted by the RNN (central panel).  The particular results shown correspond to an input soliton number $N=6$.  One can see the overall excellent visual agreement between the propagation dynamic predicted by the RNN and those simulated from the NLSE. Also notice that the distance of maximum compression and associated temporal intensity profile is particularly well predicted by the RNN. The right panel shows the relative difference between the NLSE and RNN evolution maps, with a root mean square (RMS) error computed over the full evolution $R=0.04$ (see Methods). Comparisons between NLSE and RNN evolution for 100 different input condition spanning the full range of parameter variation showed similar results with a RMS error computed over the 100 evolution maps $R=0.097$ (see Methods).\n\nA more detailed comparison between the NLSE simulations, RNN prediction, and experimental measurements at selected distances is plotted in Fig.~\\ref{fig:psexpt}. Note that in this case, third-order dispersion was also included in the training simulations (see Methods). The figure shows the intensity profiles predicted by the RNN (solid blue line), the profiles from the NLSE simulations (dashed red) as well as the experimental measurements (black dots) previously reported in \\cite{tikan2017universality}. One can see remarkable agreement at all distances between the three sets of results, and we stress particularly that the RNN reproduces both the compressed central portion and the side lobes of the Peregrine soliton associated with maximal compression around 10~m.  \n\n\n\\begin{figure}[!ht]\n  \\centering\n  \\includegraphics[width=\\linewidth]{PS_expts_v2.pdf}\n \n\\caption{Higher-order soliton ($N=6$) temporal intensity at selected distances predicted by the neural network (solid blue line), simulated with the NLSE (dashed red line), and experimentally measured (black dots). Experimental data from Ref. \\cite{tikan2017universality}.}\n\\label{fig:psexpt}\n\\end{figure}\n\nWe also tested the ability of the RNN to predict the  propagation dynamics in the spectral domain from the corresponding input spectrum. Here we use the same ensemble of NLSE numerical simulations as for the temporal evolution, but this time we train the network by feeding the spectral intensity evolution. Results for input conditions identical to that of Figs.~\\ref{fig:pstime} and \\ref{fig:psexpt} are shown in Fig.~\\ref{fig:psspec}. For convenient visualization, the evolution is plotted in logarithmic scale. The spectral evolution consists of an initial stage of spectral broadening dominated by self-phase modulation and corresponding to the compression observed in the time-domain. After the point of maximum expansion, we see a breathing phase of narrowing and re-expansion typical of higher-order soliton propagation. One can see excellent agreement between the dynamics predicted from the network and that simulated with the NLSE, with a relative discrepancy within a few dB over the entire evolution (RMS error computed over the full spectral evolution $R=0.106$). \n\nThe excellent correspondence is confirmed in Fig.~\\ref{fig:psexpt_spec} when plotting detailed comparison between the RNN predicted (blue), simulated (dashed red), and experimentally measured spectra (black dots) at selected distances around the maximal temporal compression point as previously considered and which is naturally also the point of maximum spectral broadening. In particular, one can see the excellent agreement between the NLSE and RNN results over a 25~dB dynamic range. We also performed a series of tests for 100 different input pulse spectra spanning the full range of parameter variation and found similar network performances in terms of predicted evolution with a RMS error $R=0.161$ (computed over the 100 evolution maps tested).\n\n\n\n\n\n\\begin{figure}[!ht]\n  \\centering\n  \\includegraphics[width=\\linewidth]{PS_NLSE_spec_v4.pdf}\n \n\\caption{Spectral intensity evolution of a 1.1~ps (full width at half maximum) pulse with 26.3~W peak power corresponding to an $N=6$ soliton injected into the anomalous dispersion regime of a 13~m long highly nonlinear fibre. The panels shows the result of numerical simulation (left), RNN prediction (middle), and relative difference (right). The RNN predictions use only the injected pulse spectrum as input.}\n\\label{fig:psspec}\n\\end{figure}\n\n\n\\begin{figure}[!ht]\n  \\centering\n  \\includegraphics[width=\\linewidth]{PS_expts_spec_v2.pdf}\n \n\\caption{Higher-order soliton ($N=6$) spectral intensity at selected distances predicted by the neural network (solid blue line), simulated with the NLSE (dashed red line), and experimentally measured (black dots).}\n\\label{fig:psexpt_spec}\n\\end{figure}\n\n\n\\subsection{Supercontinuum generation}\nWe next extended our study to even more complex propagation dynamics and the generation of a broadband supercontinuum. Here, we focus our attention to SC generated by injecting femtosecond pulses into the anomalous dispersion of a highly nonlinear fibre. This regime is of particular significance as it has been shown to be associated with high spectral coherence and the generation of stable frequency combs as well as to yield the broadest SC spectra \\cite{dudley2006supercontinuum}.\n\nIn order to test whether a recurrent neural network could learn SC generation dynamics and model their evolution, we generated an ensemble of SC propagation dynamics using the generalized NLSE (GNLSE) that includes the frequency-dependence of dispersion and nonlinearity, and the delayed Raman response \\cite{dudley2006supercontinuum}. Specifically, we simulated the propagation of 100~fs transform-limited pulses at 810~nm injected into the anomalous dispersion regime of a 20~cm long photonic crystal fibre with zero-dispersion at 750~nm similar to that used in \\cite{narhi2018machine}. See Methods for detailed parameter values. The ensemble includes simulations for a transform-limited input pulse with peak power uniformly distributed in the 500~W to 2~kW range that yields SC spectra with different characteristics, from isolated dispersive wave generation to fully developed octave-spanning SC with very fine spectral features. We emphasize that although the input pulse duration was kept constant for all the simulations, predicted results for other durations show similar agreement with the GNLSE as the specific cases discussed below.  \n\nWe begin by training the network from the temporal intensity evolution. Similarly to the higher-order soliton compression case, the simulation profiles are downsampled along both the propagation direction, and the temporal and spectral dimensions (see Methods). After training, the RNN model is tested for an input peak power not used in the training stage and the predicted evolution is compared with that directly simulated with the GNLSE for the same input power.  \n\n\n\\begin{figure}[!ht]\n  \\centering\n  \\includegraphics[width=\\linewidth]{SC_time_v3.pdf}\n\\caption{Temporal evolution of supercontinuum. The left panel shows the\nnumerical simulation (GNLSE) of a supercontinuum evolution in a 20~cm photonic crystal fibre for 100~fs pulse with peak power of 630~W (a) and 1.96~kW (b). See Methods for a full description of the fibre parameters. The middle panel shows the predicted (RNN) temporal intensity evolution for the same initial temporal intensity profile as in the GNLSE simulations. The right panel shows the comparison between the predicted (solid blue line) and simulated (dashed red line) profiles at selected  distances indicated by white dashed lines.}\n\\label{fig:sc_time}\n\\end{figure}\n\nResults are shown in Fig.~\\ref{fig:sc_time}(a) and (b) for an input peak power of 630~W and 1.96~kW corresponding to an input soliton number of $N=4.6$ and $N=8.1$, respectively. These values were chosen as they lead to SC with very distinct characteristics. The left panel shows the temporal intensity evolution from the GNLSE simulation and the central panel shows the predicted evolution by the RNN. The SC generation process arises from soliton dynamics including higher-order soliton compression, soliton fission and dispersive waves emission on the short wavelength side \\cite{dudley2006supercontinuum}. For longer propagation distances, solitons emerging from the fission experience the Raman self-frequency shift expanding the SC spectrum towards the long wavelengths side \\cite{dudley2006supercontinuum}. Significantly, in both scenarios, one can see the excellent visual agreement between the GNLSE simulations and RNN model. The point of soliton fission and dispersive emission as well as the red-shifting solitons parabolic trajectories are perfectly reproduced by the network. Quantitatively, the relative difference remains within a few dBs over the entire evolution down to the -30~dB bandwidth. The RMS error calculated over the full intensity evolution is $R=0.097$ and $R=0.049$ for Figs.~\\ref{fig:sc_time}(a) and (b), respectively. The remarkable ability of the RNN to predict very complex nonlinear dynamics is further highlighted in the right panel where we plot detailed comparison between the predicted and simulated SC temporal intensity at selected distances along the propagation where we can see how the amplitude and delay of the dispersive waves and Raman-shifted solitons are also predicted with excellent accuracy at all stages of the propagation. Additional predictions ran for 50 different values of pulse peak power (not used in the training phase) showed also very good agreement with the GNLSE simulations (RMS error $R=0.176$ computed over 50 different evolution maps tested).\n\n\n\n\\begin{figure}[!ht]\n  \\centering\n  \\includegraphics[width=\\linewidth]{SC_v3.pdf}\n\\caption{Spectral evolution of supercontinuum. The left panel shows the\nnumerical simulation (GNLSE) of a supercontinuum evolution in a 20~cm photonic crystal fibre for 100~fs pulse with peak power of 630~W (a) and 1.96~kW (b). See Methods for a full descritpion of the fibre parameters. The middle panel shows the predicted (RNN) temporal intensity evolution for the same initial spectral intensity profile as in the GNLSE simulations. The right panel shows the comparison between the predicted (solid blue line) and simulated (dashed red line) profiles at selected  distances indicated by white dashed lines.}\n\\label{fig:sc}\n\\end{figure}\n\nWe then tested the ability of the RNN model to predict the SC spectral intensity evolution from the input pulse spectrum. The results for an input peak power of 630~W and 1.96~kW are shown in Fig.~\\ref{fig:sc}(a) and (b), respectively. For convenient visualization, the evolution is plotted in logarithmic scale. In the case of lower peak power, one can see that the SC spectrum at the fibre output essentially consists of an isolated dispersive wave and solitons with a limited amount of red-shift. For larger input peak power, we see multiple dispersive wave emission and well-separated Raman-shifted solitons resulting in an octave-spanning SC.  Again, we can see very good visual agreement between the simulated and predicted evolution maps and that all spectral features including dispersive waves, Raman-shifted solitons and their interference that lead to fine spectral features are perfectly reproduced by the RNN. Additional predictions ran for 50 different input pulse peak power (not used in the training phase) showed also very good agreement with the GNLSE simulations (RMS error $R=0.12$).\n\n\n\n\n\\section{Conclusion}\nWe have shown that machine learning techniques can bring new insight into the study and prediction of nonlinear optical systems. Specifically, we have demonstrated that a recurrent neural network with long short-term memory can learn the complex dynamics associated with the nonlinear propagation of short pulses in optical fibres inclduing higher-order soliton compression and supercontinuum generation using solely the pulse intensity profile as input condition.  The network is also able to reproduce the dynamics both in the temporal and spectral domain, and for the particular case of higher-order soliton compression we have been able to confirm that the predicted evolutions maps are also in excellent agreement with experiments. Our results are particularly significant as applications of machine learning to ultrafast dynamics have previously been restricted to slow genetic algorithms or feed-forward neural networks designed to establish the transfer function between specific input-output parameters \\cite{farfan2018femtosecond,finot2018nonlinear,andral2015fiber,kokhanovskiy2019machine,kokhanovskiy2019machine2}. \n\nFrom an application point of view, we expect that neural networks will very soon become an important and standard tool for analysing complex ultrafast dynamics, for optimizing the generation of broadband spectra and frequency combs, as well as for designing ultrafast optics experiments. Future steps may expand the parameter space of the RNN operation by including additional training variables as e.g. the nonlinear fibre parameters. The evolution prediction may be extended to the complex field (amplitude and phase), and one could also envisage to use reverse-engineering in order to optimize the pump pulse characteristics for the generation of on-demand temporal and spectral intensity profiles at the fibre (or waveguide) output. From a more fundamental perspective, we believe that the use of recurrent neural networks will impact on future design and analysis of nonlinear physics experiments as they represent a natural candidate for exploring and analyzing complex operation regimes with long-term dependencies.  \n\n\\section*{Acknowledgements}\nLS acknowledges the Faculty of Engineering and Natural Sciences graduate school of Tampere University.\nJD acknowledges the French Agence Nationale de la Recherche (ANR-15-IDEX- 0003, ANR-17-EURE-0002).\nGG acknowledges the Academy of Finland (298463, 318082, Flagship PREIN 320165).\nThe authors also thank D. Brunner for useful discussions.\n\n\\section*{Methods}\n\n\\subsection*{Numerical simulations}\nThe numerical simulations in this work are from the NLSE and its generalized extension (1+1D) that describe the propagation of the slowly-varying optical field envelope.\n\n\\textbf{Higher-order soliton compression.}\nWe model the propagation of short pulses in the anomalous dispersion regime of a 13~m nonlinear optical fibre. The pulses have a hyperbolic-secant intensity profile centered at 1550~nm with pulse duration and peak power varying from 0.77 to 1.43~ps and from 18.41 to 34.19~W, respectively. The nonlinear coefficient of the fibre is $\\gamma = 18.4 \\times 10^{-3}$~W$^{-1}$m$^{-1}$, and the group-velocity dispersion coefficient at 1550~nm is $\\beta_2 = -5.23 \\times 10^{-27}$~s$^2$m$^{-1}$. When comparing with the experiments, third order dispersion ($\\beta_3 = 4.27 \\times 10^{-41}$~s$^3$m$^{-1}$) was also included in the training in addition to a small input pulse asymmetry caused by the experimental implementation \\cite{tikan2017universality}. The simulations use 1024 spectral\/temporal grid points with temporal window size of 10~ps and a step size of 0.13 mm (10,000 steps). For completeness, shot noise is added via one-photon-per-mode with random phase in the frequency domain, although noise effects were found to play no significant physical role in the regime of coherent propagation studied here.\n\n\\textbf{Supercontinuum generation.}\nWe model the propagation of sech-type pulse centered at 810~nm and with pulse duration of 100~fs. The peak power of the input pulse is randomly varied in range of 0.5-2~kW. The pulses are injected in the anomalous dispersion regime of a 20~cm nonlinear optical fibre, including higher-order dispersion terms, self-steepening and Raman effect. The nonlinear coefficient of the fibre is $\\gamma = 0.1$~W$^{-1}$m$^{-1}$, and the Taylor-series expansion coefficients of the dispersion at 810~nm are \n$\\beta_2 = -9.59 \\times 10^{-27}$~s$^2$m$^{-1}$,\n$\\beta_3 = 7.84 \\times 10^{-41}$~s$^3$m$^{-1}$,\n$\\beta_4 = -6.84 \\times 10^{-56}$~s$^4$m$^{-1}$,\n$\\beta_5 = -4.78 \\times 10^{-70}$~s$^5$m$^{-1}$,\n$\\beta_6 = 2.71 \\times 10^{-84}$~s$^6$m$^{-1}$ and\n$\\beta_7 = -5.00 \\times 10^{-99}$~s$^7$m$^{-1}$.\nThe simulations use 2048 spectral\/temporal grid points with temporal window size of 5~ps and a step size of 0.02 mm (10,000 steps). Shot noise is added via one-photon-per-mode with random phase in the frequency domain, but in the coherent propagation regime studied here noise effects were found to play no significant physical role.\n\n\\subsection*{Recurrent neural networks}\n\\textbf{LSTM network operation}\nThe operation of an LSTM cell can be described at time step $t$ with input $\\mathbf{x}_t \\in \\mathbb{R}^{d_o}$ by a set of equations given by \\cite{hochreiter1997long}\n\\begin{equation}\n  \\begin{split}\n    \\mathbf{f}_t &= \\sigma (\\mathbf{W}_f[\\mathbf{h}_{t-1}, \\mathbf{x}_t] + \\mathbf{b}_f)\\\\\n    \\mathbf{\\tilde{c}}_t &= \\text{tanh} (\\mathbf{W}_c[\\mathbf{h}_{t-1}, \\mathbf{x}_t] + \\mathbf{b}_c)\\\\\n    \\mathbf{o}_t &= \\sigma (\\mathbf{W}_o[\\mathbf{h}_{t-1}, \\mathbf{x}_t] + \\mathbf{b}_o)\n  \\end{split}\n\\qquad\n  \\begin{split}\n    \\mathbf{i}_t &= \\sigma (\\mathbf{W}_i[\\mathbf{h}_{t-1}, \\mathbf{x}_t] + \\mathbf{b}_i)\\\\\n    \\mathbf{c}_t &= \\mathbf{f}_t \\odot \\mathbf{c}_{t-1} + \\mathbf{i}_t \\odot \\mathbf{\\tilde{c}}_t\\\\\n    \\mathbf{h}_t &= \\mathbf{o}_t \\odot \\text{tanh}(\\mathbf{c}_t),\n  \\end{split}\n\\end{equation}\nwhere $\\mathbf{f}_t$, $\\mathbf{i}_t$ and $\\mathbf{o}_t \\in \\mathbb{R}^{d_h}$ are the forget, input and output gate vectors, respectively, with $d_h$ denoting the dimensionality of the hidden state (i.e. the number of hidden units).\nVectors $\\mathbf{c}_t$ and $\\mathbf{h}_t \\in \\mathbb{R}^{d_h}$ are the updated cell and hidden state, respectively, and $\\mathbf{W}_f$, $\\mathbf{W}_i$, $\\mathbf{W}_c$ and $\\mathbf{W}_o \\in \\mathbb{R}^{d_h \\times (d_h+d_o)}$ represent the cell weights and $\\mathbf{b}_f$, $\\mathbf{b}_i$, $\\mathbf{b}_c$ and $\\mathbf{b}_o \\in \\mathbb{R}^{d_h}$ are the biases. The sign $\\odot$ denotes point-wise multiplication. The weights and biases of the network are iteratively trained via backpropagation \\cite{werbos1990backpropagation}.\n\n\\textbf{Feed-forward network operation}\nThe operation of the fully-connected layers is similar to that in Ref.~\\cite{narhi2018machine}. The codes were written in Python using Keras \\cite{chollet2015keras} with Tensorflow backend \\cite{abadi2016tensorflow}.\n\n\\textbf{Comparison between RNN prediction and (G)NLSE simulations}\nA quantitative comparison between the network predicted evolution map and that simulated with the (G)NLSE can be performed using the average (normalized) root mean squared (RMS) errors as a metric:\n\\begin{equation}\n    \\label{eq:error}\n    R = \\sqrt{\\frac{\\sum_{i,d} (x_{m,i,d} - \\hat{x}_{m,i,d})^2}{\\sum_{i,d} (x_{m,i,d})^2}},\n\\end{equation}\nwhere $\\mathbf{x}_m$ and $\\hat{\\mathbf{x}}_m$ denote the (G)NLSE and RNN predicted intensity profile for realization $m$. The variables $i$ and $d$ indicate summation over the intensity profiles and propagation steps, respectively. When evaluating the performance of the prediction over an ensemble of $M$ evolution maps, the RMS error is calculated over $M$ distinct realizations.\n\n\\textbf{Higher-order soliton compression.}\nAn ensemble of 3,000 numerical simulations was generated. 2,900 realizations are used for the training of the RNN and 100 unseen realizations are used for testing. The simulated intensity evolution maps are uniformly downsampled at a constant propagation step of $\\Delta z = 0.13~$m, yielding 101 intensity profiles along propagation for each simulated evolution map. At every of the 101 steps, the intensity profile is convolved and downsampled with a 10~fs full width at half maximum super-Gaussian temporal filter corresponding to 145 equally spaced bins in the $\\left[-0.7,+0.7\\right]$~ps time interval. The spectral intensity profiles are convolved and downsampled with a 2~nm full width at half maximum super-Gaussian spectral filter resulting in 126 equally spaced intensity bins spanning from 1425 to 1675~nm. The temporal and spectral intensity profiles are normalized by the peak intensity over all realizations. We emphasize that because from an experimental viewpoint intensity profiles (spectral or temporal) are more straightforward to measure that the full field, we choose to only use transform-limited intensity profiles during the RNN training while the phase information is completely omitted. Both for the temporal and spectral evolution, the network is trained with intensity profiles in linear scale.  When comparing with the experiments, to account for the slight input pulse asymmetry the NLSE simulated intensity profiles of every map used in the training phase of the RNN were convolved and downsampled with a 10~fs full width at half maximum super-Gaussian temporal filter corresponding to 151 equally spaced bins in the $\\left[-0.62,+0.85\\right]$~ps time interval. The spectral intensity profiles were convolved and downsampled similarly to the case of ideal higher-order soliton propagation but spanning from 1450 to 1700~nm.\n\nThe LSTM and two hidden layers consist of 161 nodes each with ReLU activations $f(x) = \\text{max}(0,x)$, and the output layer consists of 151 and 126 nodes for temporal and spectral predictions, respectively, with sigmoid activation $f(x) = 1\/(1+\\text{exp}(-x)$. The network is trained for 60 and 120 epochs with RMSprop optimizer \\cite{tieleman2012lecture} and adaptive learning rate for the temporal and spectral intensity predictions, respectively. \n\nThe input of the RNN consists of ten consecutive temporal or spectral intensity profiles $h_{z-10\\Delta z}..h_{z-\\Delta z}$ at distance along the fibre $z-10\\Delta z$ to $z-\\Delta z$. \n\nA smaller number of intensity profiles was also found to give satisfactory results but of course this is at the expense of the relative prediction error which increases from 0.097 to 0.174 (temporal intensity evolution of higher-order soliton) when reducing the number of consecutive intensity profiles from ten to five. As the number of consecutive intensity profiles used in the training is increased, the training time also increases and therefore the training process is always a compromise between the prediction accuracy and the time required to train the network.\n\n\n\\textbf{Supercontinuum generation.}\nAn ensemble of 1,300 numerical simulations was generated. 1,250 realizations were used for training the RNN and 50 realizations for testing. The simulated intensity evolution maps are uniformly downsampled at a constant propagation step of $\\Delta z = 0.2~$mm, yielding 200 intensity profiles along propagation for each simulated evolution. In order to reduce the computational load, when training the RNN to predict temporal intensity maps, the profiles at each of the 200 steps are convolved and downsampled with a 10~fs full width at half maximum super-Gaussian temporal filter corresponding to 276 equally spaced bins spanning in the $\\left[-0.18,+1.16\\right]$~ps time interval. Note that the asymmetry in the modeled time interval is implemented to account for the soliton self-frequency shift effect. When training the RNN from spectral intensity profiles, each  spectrum is convolved and downsampled with a 2~nm full width at half maximum super-Gaussian spectral filter such that the wavelength grid consisted of 251 spectral intensity bins spanning from 550 to 1050~nm. The profiles are normalized by the peak intensity over all realizations.\n\n\nFor the temporal intensity evolution, the LSTM and two hidden layers consist of 300 nodes each with ReLU activations, and the output layer consists of 276 nodes with sigmoid activation. The network was trained for 120 epochs with RMSprop optimizer and adaptive learning rate. For the spectral intensity evolution, the LSTM and two hidden layers consist of 250 nodes each with ReLU activations, and the output layer consists of 251 nodes with sigmoid activation. The network is trained for 100 epochs. \n\n\n\n\\bibliographystyle{ieeetr}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{introducao}\n\nBefore talking about the subject of this paper, we present two topics related to this work.\nWe don't use the second topic here. It will only illustrate our main result (Theorem \\ref{teorema principal}).\n\nThe first topic is invariant metrics on homogeneous spaces. Let $M$ be a connected differentiable manifold endowed with a completely nonholonomic distribution $\\mathcal D$. \nAn absolutely continuous path $\\gamma$ in $M$ is horizontal if $\\gamma^\\prime(t) \\in \\mathcal D$ for almost every $t$.\nChow-Rashevskii theorem states that every pair of points in $M$ can be connected by a  horizontal curve (see \\cite{Chow}, \\cite{Montgomery}, \\cite{Rashevskii}).\nWe endow each subspace $\\mathcal D_x$ of $\\mathcal D$ with a norm $F(x,\\cdot)$ such that $x \\mapsto F(x,\\cdot)$ is continuous, that is, given a horizontal continuous vector field $Y$ on $M$, then $x \\mapsto F(x,Y(x))$ is a continuous map (see \\cite{Berestovskii1}).\nNow we proceed as in the definition of sub-Riemannian geometry: \nThe $C^0$-Carnot-Carath\\'eodory-Finsler metric on $M$ is given by\n\\begin{equation}\n\\label{equacao carnot caratheodory}\nd_c(x,y)=\\inf_{\\gamma \\in \\mathcal H_{x,y}}\\int_I F(\\gamma(t),\\gamma^\\prime(t))dt,\n\\end{equation}\nwhere $\\mathcal H_{x,y}$ is the set of horizontal curves connecting $x$ and $y$ (see \\cite{Berestovskii1}, \\cite{Berestovskii2}).\n\nThe following theorem due to Berestovskii is important for this work and states that if an intrinsic metric is homogeneous, then it has the tendency to gain extra regularity.\n\n\\begin{theorem}\n\\label{Berestovskii theorem}[Berestovskii, \\cite{Berestovskii2}, Theorem 3]\nIf $(M,d)$ is locally compact, locally contractible homogeneous space, endowed with an invariant intrinsic metric $d$, then $(M,d)$ is isometric to the quotient space $G\/H$ of a Lie group $G$ over a compact subgroup $H$ of it, endowed with a $G$-invariant $C^0$-Carnot-Carath\\'eodory-Finsler metric.\n\\end{theorem}\n\nThe second topic is geometric flows in Riemannian geometry. \nThere are several of them such as the mean curvature flow (for hypersurfaces in Riemannian manifolds), the Ricci-Hamilton flow, etc. \nThese flows, in some specific situations, converges to a more ``homogeneous'' Riemannian metric (eventually after some normalization of the volume).\nFor instance, the normalized Ricci-Hamilton flow converges to a Riemannian metric of constant sectional curvature for $3$-dimensional closed Riemannian manifolds with strictly positive Ricci curvature (see \\cite{Hamilton1}).\nIt is an example of a dynamical system of metrics converging to a ``well behaved'' metric.\n\nNow we give an outline of this work.\n\nLet $\\varphi: G \\times M \\rightarrow M$ be a left action of a Lie group on a differentiable manifold endowed with a metric $d$ (distance function) compatible with the topology of $M$.\nAs usual, we denote $gx:=\\varphi(g,x)$. \nLet $X$ be a compact subset of $M$.\nThe isotropy subgroup of $X$ is a closed subgroup $H_X$ of $G$ given by\n\\[\nH_X=\\{g\\in G; gX=X\\}.\n\\]\nThen there exist a unique differentiable structure on $G\/H_X$, compatible with the quotient topology, such that the natural action $\\phi: G \\times G\/H_X \\rightarrow G\/H_X$, given by $\\phi(g,hH_X)=ghH_X$, is smooth.\nIn \\cite{Benetti} and \\cite{BenettiFukuoka}, the authors define a metric $d_X$ on $G\/H_X$, which is called induced Haudorff metric, by \n\\[\nd_X(gH_X,hH_X)=d_H(gX,hX),\n\\]\nwhere $d_H$ is the Hausdorff distance in $(M,d)$ and $gH_X$ denote the left coset of $g$ in $G\/H_X$.\nSuppose that $\\varphi$ is transitive and that there exist $p\\in M$ such that $H_p=H_X$.\nThen the map $\\eta: G\/H_X \\rightarrow M$ given by $gH_X \\mapsto g.p$ is a diffeomorphism such that $\\varphi(g,\\eta(hH_X))= \\eta(\\phi(g,hH_X))$, that is, $\\varphi$ can be identified with $\\phi$.\nWe use the induced Hausdorff metric in order to create a discrete dynamical system of metrics on $M$.\nDefine $d^1=\\hat d_X$, where the hat denotes the intrinsic metric induced by $d_X$.\nWe can iterate this process in $\\varphi:G\\times (M,d^1)\\rightarrow (M,d^1)$ in order to obtain $d^2$, where $(G\/H_X,d^1)$ is identified with $(M,d^1)$ via $\\eta$.\nThrough this iteration, we can define a sequence of metrics $d,d^1,d^2, \\ldots$ on $M$, which we call {\\it sequence of induced Hausdorff metrics}.\n\nIn this work we study the sequence of induced Hausdorff metrics in the following case:\n$M$ is the Lie group $G$ itself endowed with a metric $d$ that is bounded above by a right invariant intrinsic metric $\\mathbf d$, $\\varphi: G \\times (G,d) \\rightarrow (G,d)$ is the product of $G$ and\n$X$ is a finite subset of $G$ containing the identity element $e$ (It is implicit here that $H_X=\\{e\\}$).\nWe prove that $d^i$ converges pointwise to a metric $d^\\infty$.\nMoreover if the semigroup $S_X$ generated by $X$ is dense in $G$ and $d$ is complete, then $d^\\infty$ is the distance function of a right invariant $C^0$-Carnot-Carath\\'eodory-Finsler metric.\nWe give a necessary and sufficient condition in order to $d^\\infty$ be $C^0$-Finsler. \nIn this case, an explicit formula for the Finsler metric is obtained.\n\nOf course the comparison of our work with the Riemannian geometric flows can't be taken so literally. \nIt is given here only to illustrate the dynamics of the sequence of induced Hausdorff metrics, which in some cases converges to a more ``well behaved'' metric. \nOne important restriction is that it is defined only on left coset manifolds. On the other hand the metric $d$ is much more general than Riemannian metrics and the induced Hausdorff distance is the basic tool in order to overcome the lack of differentiability.\n\nThis work is organized as follows. \nIn Section \\ref{preliminares} we fix notations and we present definitions and results that are necessary for this work. \nIn Section \\ref{sequencia de metricas} we study some properties of the sequence of induced Hausdorff metrics.\nIn particular, we prove that it is increasing, and if $d$ is bounded above by a right invariant intrinsic metric, then it converges pointwise to a metric.\nThis limit metric is intrinsic if $d$ is complete.\nIn Section \\ref{existencia sx denso}, we prove that every connected Lie group $G$ admits a finite subset $X \\ni e$ such that $H_X=\\{e\\}$ and $\\bar S_X = G$.\nIn Section \\ref{Finlser fomula}, we study the case $\\bar S_X=G$, where $d$ is complete and bounded above by a intrinsic right invariant metric. \nWe prove that the sequence of induced Hausdorff metrics converges to a right invariant $C^0$-Carnot-Carath\\'eodory-Finsler metric. \nThe case where $d^\\infty$ is $C^0$-Finsler is also studied here, as explained before.\nIn Section \\ref{secao exemplos}, we give some additional examples in order to illustrate better this work.\nFinally Section \\ref{secao final} is devoted to final remarks.\n\nThe authors would like to thank Professor Luiz A. B. San Martin for some valuable suggestions.\n\n\\section{Preliminaries}\n\\label{preliminares}\n\nIn this section we present some definitions and results that are used in this work. They can be found in \\cite{Burago}, \\cite{BenettiFukuoka}, \\cite{Helgason}, \\cite{Jurdjevic},  \\cite{KobayashiNomizu1}, \\cite{KobayashiNomizu2} and \\cite{Warner}.\nFor the sake of clearness, we usually don't give the definitions and results in their most general case.\n\nLet $G$ be a group and $(M,d)$ be a metric space. \nConsider a left action $\\varphi:G\\times M \\rightarrow M$ of $G$ on $M$.\nThen $ex=x$ for every $x\\in M$ and $(gh)x=g(hx)$ for every $(g,h,x)\\in G\\times G\\times M$. \nEvery $\\varphi_g:=\\varphi(g,\\cdot)$ is a bijection. \nWe say that a left action $\\varphi:G \\times M \\rightarrow M$ is an action by isometries if every $\\varphi_g$ is an isometry. \nAnalogously we say that $\\varphi$ is an action by homeomorphism if $\\varphi_g$ is a homeomorphism for every $g\\in G$. \n\nLet $(M,d)$ be a metric space. \nWe denote the open ball with center $p$ and radius $r$ in $(M,d)$ by $B_d(p,r)$. \nThe closed ball is denoted by $B_d[p,r]$. \nThe topology induced by $d$ is denoted by $\\tau_d$.\nThe closure of a subset $A$ in $(M,d)$ is denoted by $\\bar A$. When more than one metric or topology are involved, for instance we have metrics $d$ and $\\rho$ and a topology $\\tau$ on $M$, we use terms like $d$-neighborhood, $\\tau$-open subset, $\\rho$-compact, etc.\n\nLet $A,B$ be compact subsets of $(M,d)$. The Hausdorff distance between $A,B$ is given by\n\\[\nd_H(A,B)=\\max\\left\\{\\sup_{x\\in A} \\inf_{y \\in B} d(x,y), \\sup_{y\\in B} \\inf_{x \\in A} d(x,y) \\right\\}.\n\\]\nIt is well known that $d_H$ is a metric on the family of compact subsets of $M$.\n\nLet $\\varphi:G \\times M \\rightarrow M$ be a left action by homeomorphisms of a group $G$ on a metric space $(M,d)$.\nIf $X\\subset M$ is a subset, then the isotropy subgroup of $G$ with respect to $X$ is defined by $H_X=\\{g\\in G;gX=X\\}$.\nSuppose that $X$ is a compact subset in $M$.\nIn Proposition 2.1 of \\cite{BenettiFukuoka} (and in \\cite{Benetti}), we define the induced Hausdorff metric on the left coset space $G\/H_X$ as $d_X(gH_X,hH_X)=d_H(gX,hX)$, where $d_H$ is the Hausdorff distance in $M$.\n\nA partition $\\mathcal{P}$ of an interval $[a,b]$ is a subset $\\{ t_0,\\ldots,t_{n_{\\mathcal P}}\\} \\subset [a,b]$ such that $a=t_0 < t_1 < \\ldots < t_{n_{\\mathcal P}} = b$. \nThe norm of $\\mathcal P$ is defined as $\\vert \\mathcal P\\vert=\\max_{i=1,\\ldots,n_{\\mathcal P}} \\vert t_i - t_{i-1}\\vert$.\nThe length of a path $\\gamma:[a,b] \\rightarrow M$ on a metric space $(M,d)$ is given by\n\\[\n\\ell_d(\\gamma)=\\sup_{\\mathcal{P}}\\sum_{i=1}^{n_{\\mathcal{P}}}d(\\gamma(t_i),\\gamma(t_{i-1})).\n\\]\nIt is well known that for every $\\varepsilon >0$, there exist a $\\delta >0$ such that\n\\[\n\\ell_d(\\gamma)\\leq \\sum_{i=1}^{n_{\\mathcal P}} d(\\gamma(t_i),\\gamma(t_{i-1}))+\\varepsilon\n\\]\nfor every partition $\\mathcal P$ such that $\\vert \\mathcal{P}\\vert < \\delta$ (see \\cite{Burago}).\n\nGiven a metric space $(M,d)$ we can define the extended metric (the distance can be $\\infty$)\n\\[\n\\hat d(x,y)=\\inf_{\\gamma \\in \\mathcal{C}_{x,y}}\\ell_d(\\gamma),\n\\]\non $M$, where $\\mathcal{C}_{x,y}$ is the family of paths on $(M,d)$ that connects $x$ and $y$. \nWe denote $\\mathcal C^d_{x,y}$ instead of $\\mathcal C_{x,y}$ if there exist more than one metric defined on $M$.\n$\\hat d$ is the intrinsic (extended) metric induced by $d$, we always have that $d\\leq \\hat d$. \nWe say that the metric $d$ is intrinsic if $\\hat d=d$ and we have that a metric $\\hat d$ is always intrinsic.\n\n\\begin{proposition}\n\\label{comprimentos iguais}\nLet $(M,d)$ be a metric space and $\\hat d$ be the intrinsic (extended) metric induced by $d$. \nThen $\\ell_{\\hat d}(\\gamma)=\\ell_d (\\gamma)$ for every rectifiable curve $\\gamma$ in $(M,d)$.\n\\end{proposition}\n\nIf $(M,d)$ is a metric space, $\\varepsilon>0$ and $x,y \\in M$, then an $\\varepsilon$-midpoint of $x$ and $y$ is a point $z\\in M$ that satisfies $\\vert 2d(x,z)-d(x,y)\\vert \\leq \\varepsilon$ and $\\vert 2d(y,z) - d(x,y) \\vert \\leq \\varepsilon$. \n\nWe have the following results about intrinsic metrics. Their proofs can be found in \\cite{Burago}.\n\n\\begin{proposition}\n\\label{existenciamidpoint}\nIf $d$ is an intrinsic metric on $M$ and $\\varepsilon> 0$,  then every $x,y \\in M$ admit an $\\varepsilon$-midpoint. \n\\end{proposition}\n\n\\begin{proposition}\n\\label{midpoint}\nLet $(M,d)$ be a complete metric space. \nIf every $x,y\\in M$ admit a $\\varepsilon$-midpoint (for every $\\varepsilon >0$), then $d$ is intrinsic.\n\\end{proposition}\n\n$C^0$-Carnot-Carath\\'eodory-Finsler-metrics are intrinsic because they come from a length structure (see \\cite{Burago}).\nA particular case of $C^0$-Carnot-Carath\\'eodory-Finsler metrics are the $C^0$-Finsler metrics.\nLet $M$ be a differentiable manifold and $TM=\\{(x,v);x\\in M, v\\in T_xM\\}$ be its tangent bundle. \nA $C^0$-Finsler metric on $M$ is a continuous function $F:TM \\rightarrow \\mathbb{R}$ such that $F(x,\\cdot)$ is a norm on $T_xM$ for every $x\\in M$. \n$F$ induces a metric $d_F$ on $M$ given by\n\\[\nd_F(x,y)=\\inf_{\\gamma \\in \\mathcal{S}_{x,y}}\\ell_F(\\gamma),\n\\]\nwhere\n\\[\n\\ell_F(\\gamma)=\\int_a^b F(\\gamma(t),\\gamma^\\prime(t))dt,\n\\]\nis the length of $\\gamma$ in $(M,F)$ and $\\mathcal{S}_{x,y}$ is the family of paths on $M$ which are smooth by parts and connects $x$ and $y$. \nHere the family $\\mathcal S_{x,y}$ can be replaced by the family of absolutely continuous paths connecting $x$ and $y$ (see \\cite{Berestovskii1}).\nIf $d:M\\times M \\rightarrow \\mathbb{R}$ is a metric on $M$, then we say that $d$ is $C^0$-Finsler if there exist a $C^0$-Finsler metric $F$ on $M$ such that $d=d_F$.\nA differentiable manifold endowed with a $C^0$-Finsler metric is a $C^0$-Finsler manifold.\n\n\\begin{remark}\n\\label{outrofinsler}\nThere are another usual (in fact more usual) definition of Finsler manifold, where $F$ satisfies other conditions  (see, for instance, \\cite{BaoChernShen} and \\cite{Deng}): $F$ is smooth on $TM-TM_0$, where $TM_0=\\{(p,0)\\in TM;p \\in M\\}$ is the zero section, and  $F(p,\\cdot)$ is a Minkowski norm on $T_pM$ for every $p\\in M$. \nIn order to make this difference clear we put the prefix $C^0$ before Finsler.\n\n$C^0$-Finsler metrics are studied, for instance, in \\cite{Berestovskii1}, \\cite{Berestovskii2}, \\cite{Burago} and \\cite{Gribanova}.\n\\end{remark}\n\n\\begin{remark}\n\\label{relacoes Carnot Caratheodory}\nLet $M$ be a connected differentiable manifold and $\\mathcal D$ be a completely nonholonomic distribution on $M$.\nLet $F$ as in the definition of the $C^0$-Carnot-Carath\\'eodory-Finsler metric in the introduction.\nObserve that there exist smooths sections $\\mathbf g_1, \\mathbf g_2$ of inner products in $\\mathcal D$ such that $\\mathbf g_1 \\leq F \\leq \\mathbf g_2$.\nIn fact, this is clear locally, and in order to see that this holds globally, just use partition of unity.\nThis implies that balls in $C^0$-Carnot-Carath\\'eodory-Finsler metrics are contained and contains a ball in some Carnot-Carath\\'eodory metric of the same distribution. \nBall-box theorem gives a qualitative behavior of balls in sub-Riemannian manifolds (see \\cite{Montgomery}) and this behavior depends only on the distribution $\\mathcal D$.\nTherefore balls in $C^0$-Carnot-Carath\\'eodory-Finsler metrics of the same distribution have the same qualitative behavior.\n\nAnother consequence of the ball-box theorem is that a Carnot-Carath\\'eodory metric correspondent to a non-trivial completely nonholonomic distribution can't the bounded above by the distance function of a Riemannian metric.\nTherefore a $C^0$-Carnot-Caratheodory-Finsler metric correspondent to a non-trivial completely nonholonomic distribution can't be bounded above by a $C^0$-Finsler metric.\n\\end{remark}\n\nThe following theorem is the version of the classical Hopf-Rinow theorem for intrinsic metrics (see \\cite{Burago}).\n\n\\begin{theorem}[Hopf-Rinow-Cohn-Vossen theorem]\n\\label{hopf rinow intrinsic}\nLet $(M,d)$ be a locally compact metric space endowed with an intrinsic metric.\nThen the following assertions are equivalent:\n\\begin{itemize}\n\\item $(M,d)$ is complete;\n\\item $B_d[p,r]$ is compact for every $p \\in M$ and $r > 0$;\n\\item Every geodesic (local minimizer parameterized by arclength) $\\gamma:[0,a) \\rightarrow M$ can be extended to a continuous path $\\bar \\gamma:[0,a] \\rightarrow M$;\n\\item There exist a $p \\in M$ such that every shortest path parameterized by arclength $\\gamma:[0,a) \\rightarrow M$ satisfying $\\gamma(0)=p$ admits a continuous extension $\\bar \\gamma:[0,a] \\rightarrow M$.\n\\end{itemize}\n\\end{theorem}\n\n\\begin{lemma}\n\\label{fechado limitado compacto}\nAny closed and bounded subset of a Lie group endowed with a right invariant intrinsic metric is compact. \nIn particular, a right invariant intrinsic metric on a Lie group is complete.\n\\end{lemma}\n\n{\\it Proof}\n\n\\\n\nA right invariant intrinsic metric $\\tilde d$ is a $C^0$-Carnot-Carath\\'eodory Finsler metric that comes from a continuous right invariant section of norms $F$ in a right invariant completely nonholonomic distribution $\\mathcal D$ (see Theorem \\ref{Berestovskii theorem}). \nObserve that $\\tilde d \\geq d_{\\mathbf g}$ for some right invariant Riemannian manifold $\\mathbf g$.\nIn fact, choose a right invariant smooth section of inner products $\\bar{\\mathbf g}$ on $\\mathcal D$ such that $\\bar{\\mathbf g} \\leq F$ and then extend $\\bar{\\mathbf g}$ to the whole tangent spaces resulting in a right invariant Riemannian metric $\\mathbf g$.\nThe completeness of $(M,\\mathbf g)$ and the classical Hopf-Rinow theorem for Riemannian manifolds implies that any closed bounded subset of $(M,\\mathbf g)$ is compact.\nTherefore any closed bounded subset of $(M,\\tilde d)$ is compact because $\\tilde d \\geq d_{\\mathbf g}$.\n\nThe last statement is due to Theorem \\ref{hopf rinow intrinsic}.$\\blacksquare$\n\n\\\n\nThe rest of this section is about the orbits of a family of vector fields and it is used in Section \\ref{existencia sx denso} in order to prove the existence of $X$ such that $S_X$ is dense in $G$ (see \\cite{Jurdjevic}).\n\n\\begin{definition}\n\\label{orbita}\nLet $\\mathcal{F}$ be a family of complete vector fields on a differentiable manifold $M$ and $x\\in M$. \nFor $X \\in \\mathcal{F}$, denote by $t\\mapsto (\\exp tX)(x)$ be the integral curve of $X$ such that $(\\exp 0X) (x)=x$.\nThe orbit $G(x)$ of $\\mathcal{F}$ through $x$ is the set of points given by $(\\exp t_mX_m)(\\exp t_{m-1}X_{m-1}) \\ldots (\\exp t_1X_1)(x)$, with $m\\in \\mathbb N$, $t_i \\in \\mathbb{R}$ and $X_i \\in \\mathcal{F}$, $i=1,\\ldots m$. \n\\end{definition}\n\nIn the conditions of Definition \\ref{orbita}, denote by $Lie(\\mathcal{F})$ the Lie algebra generated by $\\mathcal{F}$ and let $Lie_x(\\mathcal{F})$ be the restriction of $Lie(\\mathcal{F})$ to the tangent space $T_xM$.\n\n\\begin{theorem}[Hermann-Nagano Theorem]\n\\label{nagano}\nLet $M$ be an analytic manifold and $\\mathcal{F}$ be a family of analytic vector fields on $M$. Then\n\\begin{itemize}\n\\item each orbit of $\\mathcal F$ is an analytic submanifold of $M$;\n\\item if $N$ is an orbit of $\\mathcal{F}$, then the tangent space of $N$ in $x$ is given by $\\text{Lie}_x(\\mathcal F)$.\n\\end{itemize}\n\n\\end{theorem}\n\n\\begin{corollary}\n\\label{nagano para grupos de Lie}\nLet $G$ be a connected Lie group and $V$ be a vector subspace of $\\mathfrak{g}$ such that the Lie algebra generated by $V$ is $\\mathfrak{g}$.\nLet $\\{v_1,\\ldots,v_k\\}$ be a basis of $V$ and $\\mathcal{F}$ be the set of left (or right) invariant vector fields with respect to $\\{v_1,\\ldots,v_k\\}$.\nThen, for every $x \\in G$, there exist $m\\in \\mathbb{N}$, $t_{i_1},\\ldots, t_{i_m} \\in \\mathbb R$ and $v_{i_1},\\ldots, v_{i_m} \\in X$ such that $x=\\exp(t_{i_m}v_{i_m})\\ldots \\exp(t_{i_1}v_{i_1})$.\n\\end{corollary}\n\n{\\it Proof}\n\n\\\n\nIt is enough to observe that $G$ is an analytic manifold, that the right (and left) invariant vector fields with respect to $v_1,\\ldots, v_k$ are analytic (see \\cite{Helgason}) and that $\\text{Lie}_e(\\mathcal{F})=\\mathfrak g$.\nThen the orbit of $e$ is an analytic submanifold of $G$ that contains a neighborhood of $e$ (see Theorem \\ref{nagano}) and this neighborhood generates $G$ due to the connectedness of $G$.$\\blacksquare$\n\n\n\\section{Convegence of the sequence of induced Hausdorff metrics on Lie groups}\n\\label{sequencia de metricas}\n\nLet $d$ be a metric on a Lie group $G$ such that $\\tau_d=\\tau_G$, $\\varphi: G \\times (G,d) \\rightarrow (G,d)$ be the product of $G$ and $X= \\{ x_1, \\ldots, x_k \\} \\ni e$ be a finite subset of $G$ such that $H_X=\\{e\\}$.\nIn this section, we study properties of the sequence of induced Hausdorff metrics $d,d^1,d^2,\\ldots,d^i, \\ldots$ on $G$.\n\nFor every $j=1, \\ldots, k$, we define the metric $d_j(p,q)=d(px_j,qx_j)$ on $G$.\n$\\tau_{d_j}=\\tau_d$ because right translations are homeomorphisms on $G$.\nDefine also  the metric $d_M(p,q):=\\max\\limits_{j=1,\\ldots, k}d_j(p,q):=\\max\\limits_{j=1,\\ldots, k}d(px_j,qx_j)$ on $G$.\n\nFirst of all we prove that $\\tau_{d_X}=\\tau_d$ (Proposition \\ref{dedXmesmatopologia}).\n\n\\begin{lemma}\n\\label{comparadXdMd}\n$d\\leq d_M$ and $d_X \\leq d_M$.\n\\end{lemma}\n\n{\\it Proof}\n\n\\\n\nThe first inequality is obvious. The second inequality follows because\n\\[\nd_X(p,q)\n=\\max \\left\\{ \\max_i \\min_j d (px_i,qx_j), \\max_j \\min_i d (px_i,qx_j) \\right\\}\n\\]\n\\[\n\\leq \\max \\left\\{ \\max_i d (px_i,qx_i), \\max_j d (px_j,qx_j) \\right\\}\n=\\max_i d (px_i,qx_i)\n=d_M(p,q).\\blacksquare\n\\]\n\n\\begin{lemma}\n\\label{localmenteigual}\nFor every $g\\in G$, there exist a $G$-neighborhood $V$ of $g\\in G$ such that $d_X\\vert_{V \\times V}=d_M\\vert_{V \\times V}$.\n\\end{lemma}\n\n{\\it Proof}\n\n\\\n\nFor $i,j=1,\\ldots , k$, $i\\neq j$, define $\\rho_{ij}:G\\times G\\rightarrow \\mathbb R$ as $\\rho_{ij}(p,q)=d(px_i,qx_j)-\\max_k d(px_k,qx_k)$.\nThen for every $g\\in G$, there exist a $G$-neighborhood $V$ of $g$ such that \n\\[\n\\rho_{ij}(p,q)>0 \\text{ for every } p,q \\in V \\text{ and every }i\\neq j,\n\\]\nbecause $\\rho_{ij}(g,g)>0$.\nIt implies that\n\\[\nd_X(p,q)\n=\\max \\left\\{ \\max_i \\min_j d (px_i,qx_j), \\max_j \\min_i d (px_i,qx_j) \\right\\}\n\\]\n\\begin{equation}\n\\label{vizinhanca pequena}\n=\\max \\left\\{ \\max_i d (px_i,qx_i), \\max_j d (px_j,qx_j) \\right\\}\n=\\max_i d(px_i,qx_i)=d_M(p,q)\n\\end{equation}\nfor every $p,q \\in V$.$\\blacksquare$\n\n\\begin{remark}\nThe formula $d_X(p,q)=\\max_i d(px_i,qx_i)$ for every $p,q$ in a sufficiently small neighborhood of $g$ will be used several times in this work.\n\\end{remark}\n\n\\begin{lemma}\n\\label{dMdmesmatopologia} $\\tau_d = \\tau_{d_M}(=\\tau_G)$ on $G$.\n\\end{lemma}\n\n{\\it Proof}\n\n\\\n\nThe inequality $d\\leq d_M$ implies that $\\tau_d \\subset \\tau_{d_M}$.\n\nIn order to see that $\\tau_{d_M} \\subset \\tau_d$, observe that an open ball $B_{d_M}(p,r)$ can be written as\n\\[\nB_{d_M}(p,r)=\\bigcap_{j=1}^k B_{d_j}(p,r),\n\\]\nwhat implies that it is an open subset of $\\tau_d$. Therefore $\\tau_{d_M}\\subset \\tau_d$.$\\blacksquare$\n\n\\begin{lemma}\n\\label{BolasdXcompactas}\n$B_{d_X}(p,r)$ is contained in a $G$-compact subset of $G$ for a sufficiently small $r > 0$.\n\\end{lemma}\n\n{\\it Proof}\n\n\\\n\nJust observe that\n\\[\nB:=\\bigcup_{i=1}^k B_d(px_i,r)\\supset B_{d_X}(p,r).\n\\]\nIn fact, if $y \\not \\in B$, then\n\\[\n\\begin{array}{ccl}\nd_X(p,y)\n&\n=\n&\n\\max\\left\\{ \\max\\limits_i \\min\\limits_j d (px_i,yx_j), \\max\\limits_j \\min\\limits_i d (px_i,yx_j) \\right\\} \\\\\n&\n\\geq \n&\n\\max\\left\\{ \\max\\limits_i \\min\\limits_j d (px_i,yx_j), \\min\\limits_i d (px_i,y.e) \\right\\} \\\\\n&\n\\geq\n& \n\\max\\left\\{ \\max\\limits_i \\min\\limits_j d (px_i,yx_j), r \\right\\}\n\\geq r,\n\\end{array}\n\\]\nwhere the second inequality holds because $y \\not \\in B$. \nThen the result follows because $B$ is contained in a $G$-compact subset of $G$ for a sufficiently small $r$.\n\n\\begin{proposition}\n\\label{dedXmesmatopologia}\n$\\tau_d = \\tau_{d_X}$ on $G$.\n\\end{proposition}\n\n{\\it Proof}\n\n\\\n\nIt is enough to prove that $\\tau_{d_X} = \\tau_{d_M}$ due to Lemma \\ref{dMdmesmatopologia}.\n\nWe know that $d_X \\leq d_M$ (Lemma \\ref{comparadXdMd}), what implies that $\\tau_{d_X} \\subset \\tau_{d_M}$.\n\nIn order to prove that $\\tau_{d_M} \\subset \\tau_{d_X}$, we consider an open ball $B_{d_M}(p,r)$ and we prove that there exist an $\\varepsilon > 0$ such that $B_{d_X}(p,\\varepsilon)\\subset B_{d_M}(p,r)$.\nWithout loss of generality, we can consider $r > 0$ such that $B_{d_X}(p,r)$ has compact closure (see Lemma \\ref{BolasdXcompactas}).\nConsider a $G$-neighborhood $V$ of $p$ according to Lemma \\ref{localmenteigual}.\nWe can eventually consider a (further) smaller $r$ in such a way that $B_{d_M}(p,r) \\subset V$ (see Lemma \\ref{dMdmesmatopologia}). Then $B_{d_X}(p,r)\\cap V=B_{d_M}(p,r)$ due to Lemma \\ref{localmenteigual}.\n\nIf $B_{d_X}(p,r)-V$ is the empty subset, then we have that $B_{d_X}(p,r) = B_{d_M}(p,r)$ and we are done. \nOtherwise we consider\n\\[\n2\\varepsilon=\\inf\\limits_{x \\in B_{d_X}(p,r)-V}d_X(p,x).\n\\]\nObserve $d_X$ is $d \\times d$-continuous because $\\tau_{d_X}\\subset \\tau_{d}$ and $\\varepsilon$ is strictly positive because $p$ is not contained in the $G$-compact subset $\\overline{B_{d_X}(p,r)-V} $. \nTherefore\n\\[\nB_{d_X}\\left(p,\\varepsilon\\right)\n=B_{d_X}\\left(p,\\varepsilon \\right) \\cap V\n\\subset B_{d_M}(p,r)\n\\]\nwhat settles the proposition.$\\blacksquare$\n\n\\begin{definition}\n\\label{semigrupo}\nLet $X=\\{x_1,\\ldots, x_k\\}\\ni e$ be a finite subset of a Lie group $G$. The semigroup generated by $X$ is defined as \n\\[\nS_X=\\{x_{i_1}\\ldots x_{i_m};m\\in \\mathbb N, i_j \\in (1,\\ldots, k), j\\in (1,\\ldots, m)\\}.\n\\]\n\\end{definition}\n\nIn what follows, if $X=\\{x_1,\\ldots, x_k\\}$, then $X^{-1}:=\\{x_1^{-1}, \\ldots, x_k^{-1}\\}$.\n\n\\begin{proposition}\n\\label{x e menos x}\nLet $X$ be a finite subset of a Lie group $G$. \nThen $S_X$ is dense in $G$ iff $S_{X^{-1}}$ is dense in $G$.\n\\end{proposition}\n\n{\\it Proof}\n\n\\\n\nIt is enough to observe that if $i:G \\rightarrow G$ is the inversion map, then $i(S_X)=S_{X^{-1}}$.$\\blacksquare$\n\n\\\n\n\\begin{lemma}\n\\label{d1 e maior}\nLet $G$ be a Lie group and\n$d:G\\times G\\rightarrow \\mathbb R$ be a metric on $G$ such that $\\tau_d=\\tau_G$. \nLet $\\varphi: G \\times (G,d)\\rightarrow (G,d)$ be the product of $G$ and\n$X=\\{x_1,\\ldots, x_k\\}\\ni e$ be a finite subset of $G$ such that $H_X=\\{e\\}$.\nThen $d\\leq d^1$ and the sequence of induced Hausdorff metrics is increasing.\n\\end{lemma}\n\n{\\it Proof}\n\n\\\n\nLet $x,y \\in G$ and $\\gamma:[a,b]\\rightarrow G$ be a $d_X$-path (which is also a $G$-path due to Proposition \\ref{dedXmesmatopologia}) connecting $x$ and $y$ (If there isn't any path connecting $x$ and $y$, then $d(x,y)\\leq d^1(x,y) = \\infty$).\nThen\n\\[\n\\ell_{d_X}(\\gamma)=\\sup_{\\mathcal P}\\sum_{i=1}^{n_{\\mathcal P}}d_X(\\gamma(t_i),\\gamma(t_{i-1})).\n\\]\nCover $\\gamma([a,b])$ with open subsets $V$ of Lemma \\ref{localmenteigual} such that (\\ref{vizinhanca pequena}) holds. \nLet $\\varepsilon$ be the $d$-Lebesgue number of this covering. Let $\\delta >0$ such that if $\\vert t_1 - t_2\\vert < \\delta$, then $d(\\gamma(t_2),\\gamma(t_1))<\\varepsilon$.\nConsider only partitions with norm less that $\\delta$. Then\n\\begin{equation}\n\\label{calculo comprimento}\n\\ell_{d_X}(\\gamma)\n=\n\\sup_{\\mathcal P}\\sum_{i=1}^{n_{\\mathcal P}}\\max_j d(\\gamma(t_i)x_j,\\gamma(t_{i-1})x_j) \n\\geq\n\\sup_{\\mathcal P}\\sum_{i=1}^{n_{\\mathcal P}} d(\\gamma(t_i),\\gamma(t_{i-1})) = \\ell_d(\\gamma)\n\\end{equation}\ndue to (\\ref{vizinhanca pequena}).\nThen\n\\[\nd^1(x,y)\n=\\inf_{\\gamma \\in \\mathcal C_{x,y}^{d_X}} \\ell_{d_X}(\\gamma)\n\\geq \\inf_{\\gamma \\in \\mathcal C_{x,y}^d} \\ell_{d}(\\gamma)\n=\\hat d(x,y)\\geq d(x,y).\\blacksquare\n\\]\n\n\n\n\\begin{theorem}\n\\label{local maximo 1}\nLet $G$ be a Lie group and\n$d:G\\times G\\rightarrow \\mathbb R$ be a metric on $G$ such that $\\tau_d=\\tau_G$.\nLet $\\varphi: G \\times (G,d)\\rightarrow (G,d)$ be the product of $G$ and\n$X=\\{x_1,\\ldots, x_k\\}\\ni e$ be a finite subset of $G$ such that $H_X=\\{e\\}$.\nSuppose that $d$ is bounded above by a right invariant intrinsic metric $\\mathbf d$ (what implies that $G$ is connected). \nThen every $d^i$ is bounded above by $\\mathbf d$ and the sequence of induced Hausdorff metrics converges pointwise to a metric $d^\\infty$.\n\\end{theorem}\n\n{\\it Proof}\n\n\\\n\nWe prove that if $d\\leq \\mathbf d$, then $d^1 \\leq \\mathbf d$.\nThis is enough to prove that $d^i\\leq \\mathbf d$ for every $i$: just  iterate the process, replacing $d$ by $d^{i-1}$.\n\nLet $\\gamma$ be a path defined on a closed interval. Then\n\\begin{eqnarray}\n\\ell_{d_X}(\\gamma)\n&\n=\n&\n\\sup_{\\mathcal P}\\sum_{i=1}^{n_{\\mathcal P}}\\max_j d(\\gamma(t_i)x_j,\\gamma(t_{i-1})x_j) \\nonumber \\\\\n&\n\\leq\n& \n\\sup\\limits_{\\mathcal P}\\sum\\limits_{i=1}^{n_{\\mathcal P}}\\max\\limits_j \\mathbf d(\\gamma(t_i)x_j,\\gamma(t_{i-1})x_j) \\nonumber\n\\\\\n&\n=\n&\n\\sup\\limits_{\\mathcal P}\\sum\\limits_{i=1}^{n_{\\mathcal P}} \\mathbf d (\\gamma(t_i),\\gamma(t_{i-1}))= \\ell_{\\mathbf d}(\\gamma) \\label{majoracao}\n\\end{eqnarray}\nbecause $\\mathbf d$ is right invariant. \nFrom (\\ref{majoracao}) and the fact that $\\mathbf d$ is intrinsic, we have that \n\\[\nd^1(x,y) \n= \n\\inf_{\\gamma \\in \\mathcal{C}_{x,y}^{d_X}} \\ell_{d_X}(\\gamma) \n\\leq \n\\inf_{\\gamma \\in \\mathcal{C}_{x,y}^{\\mathbf d}} \\ell_{\\mathbf d}(\\gamma)\n=\n\\mathbf d(x,y)\n\\]\nfor every $x,y \\in G$ (Observe that $\\mathcal C_{x,y}^{d_X}= \\mathcal C_{x,y}^{\\mathbf d}$ due to Proposition \\ref{dedXmesmatopologia}).\n\nObserve that $d^i$ is an increasing sequence of metrics bounded above by $\\mathbf d$.\nThen $d^i$ converges pointwise to a function $d^\\infty$ and it is easy to prove that $d^\\infty$ is a metric.$\\blacksquare$\n\n\\\n\nObserve that if $d$ is a metric on a group $G$, then $\\bar d: G \\times G \\rightarrow \\mathbb R$, defined as $\\bar d(x,y)=\\sup_{\\sigma \\in G}d(x\\sigma,y\\sigma)$, is a right invariant extended metric. \nIts intrinsic extended metric $\\hat{\\bar d}(x,y)=\\inf_{{\\mathcal C}^{\\bar{d}}_{x,y}} \\ell_{\\bar d}(\\gamma)$ is also a right invariant.\n\n\\begin{corollary}\n\\label{d barra chapeu limitante} \nLet $G$ be a Lie group and\n$d:G\\times G\\rightarrow \\mathbb R$ be a metric on $G$ such that $\\tau_d=\\tau_G$.\nLet $\\varphi: G \\times (G,d)\\rightarrow (G,d)$ be the product of $G$ and\n$X=\\{x_1,\\ldots, x_k\\}\\ni e$ be a finite subset of $G$ such that $H_X=\\{e\\}$. \nSuppose that $\\hat{\\bar{d}}$ is a metric on $G$.\nThen the sequence of induced Hausdorff metrics converges to a metric $d^\\infty$ on $G$.\n\\end{corollary}\n\n{\\it Proof}\n\n\\\n\nIt is enough to see that $\\hat{\\bar{d}}$ is a right invariant intrinsic metric such that $d\\leq \\hat{\\bar{d}}$.$\\blacksquare$  \n\n\\begin{lemma}\n\\label{translacao a direita decresce}\nLet $d^i$ be a sequence of induced Hausdorff metrics converging to $d^\\infty$.\nThen $d^\\infty(x,y) \\geq d^\\infty(x \\sigma,y \\sigma)$ for every $\\sigma \\in \\bar S_X$ and every $x,y \\in G$.\nIn particular $\\ell_{d^\\infty}(\\gamma) \\geq \\ell_{d^\\infty}(\\gamma \\sigma)$ for every $\\sigma \\in \\bar S_X$ and every path $\\gamma$. \nMoreover if $\\bar S_X$ is a subgroup, then $d^\\infty$ is invariant by the right action of $\\bar S_X$. \nIn particular, if $\\bar S_X=G$, then $d^\\infty$ is right invariant.\n\\end{lemma}\n\n{\\it Proof}\n\n\\\n\nIn order to prove that $d^\\infty(x,y) \\geq d^\\infty(x\\sigma,y \\sigma)$ for every $(x,y,\\sigma) \\in G \\times G \\times \\bar S_X$, it is enough to prove that $d^\\infty(x,y) \\geq d^\\infty(x x_j,y x_j)$ for every $(x,y,x_j) \\in G \\times G \\times X$ because every $\\sigma \\in \\bar S_X$ can be arbitrarily approximated by product of elements of $X$.\n\nLet $d$ be a metric on $G$.\nThen \n\n\\begin{eqnarray}  \n\\ell_{d^1}(\\gamma) &\n= & \\sup\\limits_{\\mathcal P}\\sum\\limits_{i=1}^{n_{\\mathcal P}} d^1(\\gamma(t_i),\\gamma(t_{i-1})) \n= \n\\sup\\limits_{\\mathcal P}\\sum\\limits_{i=1}^{n_{\\mathcal P}} d_X(\\gamma(t_i),\\gamma(t_{i-1})) \\nonumber \\\\\n&\n=\n&\n\\sup\\limits_{\\mathcal P}\\sum\\limits_{i=1}^{n_{\\mathcal P}} \\max\\limits_j d(\\gamma(t_i)x_j,\\gamma(t_{i-1})x_j) \\nonumber \\\\\n& \n\\geq\n&\n\\sup\\limits_{\\mathcal P}\\sum\\limits_{i=1}^{n_{\\mathcal P}} d(\\gamma(t_i)x_j,\\gamma(t_{i-1})x_j)=\\ell_{d}(\\gamma x_j) \\label{multiplica xj diminui}\n\\end{eqnarray}\nfor every $x_j \\in X$.\nThe second equality holds due to Proposition \\ref{comprimentos iguais}.\n\nFix $(x,y,x_j) \\in G \\times G \\times X$. Observe that $d^i(x,y)\\geq d^{i-1}(x x_j, y x_j)$ for $i \\geq 2$ due to (\\ref{multiplica xj diminui}), $\\lim_{i \\rightarrow \\infty}d^i(x,y)=d^\\infty(x,y)$ and $\\lim_{i\\rightarrow \\infty} d^{i-1}(x x_j, y x_j)=d^\\infty (xx_j, yx_j)$.\nThen $d^\\infty(x,y)\\geq d^\\infty(xx_j, yx_j)$, what implies that $d^\\infty(x,y)\\geq d^\\infty(x\\sigma, y\\sigma)$ for every $\\sigma \\in \\bar S_X$.\n\nIn order to prove that if $\\bar S_X$ is a subgroup, then $d^\\infty$ is invariant by the right action of $\\bar S_X$, it is enough to observe that $d^\\infty(x,y)\\geq d^\\infty(x\\sigma, y\\sigma)\\geq d^\\infty(x\\sigma \\sigma^{-1}, y\\sigma\\sigma^{-1})=d^\\infty(x,y)$.$\\blacksquare$\n\n\\\n\nNow we prove that if $d$ is complete and bounded above by a right invariant intrinsic metric, then $d^\\infty$ is intrinsic (Theorem \\ref{completo eh intrinseco}). \n\n\\begin{lemma}\n\\label{osdoiscompletos}\nLet $M$ be a set and suppose that $d$ and $\\rho$ are metrics on $M$ such that $d \\leq \\rho$ and $\\tau_d=\\tau_\\rho$. \nIf $d$ is complete, then $\\rho $ is also complete. \n\\end{lemma}\n\n{\\it Proof}\n\n\\\n\nJust notice that every Cauchy sequence in $(M,\\rho)$ is a Cauchy sequence in $(M,d)$. \nThen the sequence converges in both metrics spaces because $\\tau_d=\\tau_\\rho$.$\\blacksquare$\n\n\\begin{lemma}\n\\label{limiteintrinseco}\nLet $M$ be locally compact topological space and suppose that $d^i$ is a sequence intrinsic metrics on $M$ that converges pointwise to metric  $d^\\infty$.\nSuppose that there exist a complete intrinsic metric $d_l$ and a metric $d_h$ on $M$ such that $\\tau_{d_l}=\\tau_{d_h}=\\tau_M$ and $d_l \\leq d^i \\leq d_h$ for every $i$.\nThen $d^\\infty$ is a complete intrinsic metric.\n\\end{lemma}\n\n{\\it Proof}\n\n\\\n\nFirst of all $d^\\infty$ is complete due to Lemma \\ref{osdoiscompletos} and because $d^\\infty \\geq d_l$. Moreover it is straightforward that $\\tau_{d^\\infty}=\\tau_M$.\n\nLet $\\varepsilon >0$ and $x,y \\in M$. \nWe will prove that $x$ and $y$ admits an $\\varepsilon$-midpoint $z$ with respect to the metric $d^\\infty$ (see Proposition \\ref{midpoint}).\n\nLet $z_i$ be an $\\varepsilon \/3$-midpoint of $x$ and $y$  with respect to the metric $d_i$ (see Proposition \\ref{existenciamidpoint}). We claim that the sequence $z_i$ is contained in a compact subset.\nIn fact \n\\[\n\\left\\vert 2d^i(x,z_i)-d^i(x,y) \\right\\vert \\leq \\frac{\\varepsilon}{3}\n\\]\nimplies that\n\\[\n2d^i(x,z_i) \n\\leq d^i(x,y) + \\frac{\\varepsilon}{3}\n\\]\nand\n\\[\n2d_l(x,z_i) \n\\leq 2d^i(x,z_i) \n\\leq d^i(x,y) + \\frac{\\varepsilon}{3} \n\\leq d_h(x,y) + \\frac{\\varepsilon}{3}.\n\\]\nThen the sequence $z_i$ is bounded in $(M,d_l)$ and it is contained in a compact subset due to Theorem \\ref{hopf rinow intrinsic}.\n\nLet $z$ be an accumulation point of $z_i$.\nWe claim that $z$ is an $\\varepsilon$-midpoint with respect to $d^\\infty$.\nFirst of all, taking a subsequence $z_i$ coverging to $z$, we claim that $d^i(x,z_i)$ converges to $d^\\infty (x,z)$ as $i$ goes to infinity.\nIn fact, for every $\\varepsilon >0$, there exist $N_1\\in \\mathbb N$ such that\n\\[\n\\vert d^\\infty(x,z) - d^i(x,z_i) \\vert \n\\leq \\vert d^\\infty(x,z) - d^i(x,z) + d^i(x,z)- d^i(x,z_i) \\vert\n\\]\n\\[\n\\leq \\vert d^\\infty(x,z) - d^i(x,z)\\vert + \\vert d_h(x,z)- d_h(x,z_i) \\vert < \\varepsilon \/3\n\\]\nfor every $i\\geq N_1$, and $d^i(x,z_i)$ converges to $d^\\infty(x,z)$. Then for every $\\varepsilon >0$, there exist a $N_2 \\in \\mathbb{N}$ such that\n\\[\n\\vert 2d^\\infty(x,z) - d^\\infty(x,y)\\vert\n\\]\n\\[\n\\leq \\left\\vert 2d^\\infty(x,z)-2d^i(x,z_i) \\vert+ \\vert 2d^i(x,z_i)-d^i(x,y) \\vert + \\vert d^i(x,y)-d^\\infty(x,y) \\right\\vert \n< \\varepsilon\n\\]\nfor every $i \\geq N_2$. The inequality \n\\[\n\\vert 2d^\\infty(y,z) - d^\\infty(x,y)\\vert < \\varepsilon\n\\]\nis analogous. Then $z$ is an $\\varepsilon$-midpoint of $x$ and $y$ in $(M,d^\\infty)$ and $d^\\infty$ is intrinsic due to Proposition \\ref{midpoint}.$\\blacksquare$\n\n\\begin{theorem}\n\\label{completo eh intrinseco}\nLet $G$ be a Lie group and\n$d:G\\times G\\rightarrow \\mathbb R$ be a complete metric on $G$ such that $\\tau_d=\\tau_G$.\nLet $\\varphi: G \\times (G,d)\\rightarrow (G,d)$ be the product of $G$ and\n$X=\\{x_1,\\ldots, x_k\\}\\ni e$ be a finite subset of $G$ such that $H_X=\\{e\\}$. \nSuppose that there exist an intrinsic right invariant metric $\\mathbf d$ on $G$ such that $d \\leq \\mathbf d$.\nThen the sequence of induced Hausdorff metrics converges to a complete and intrinsic metric $d^\\infty$. \n\\end{theorem}\n\n{\\it Proof}\n\n\\\n\nObserve that $\\tau_{\\mathbf d} = \\tau_d = \\tau_G$ because $\\mathbf d$ is $C^0$-Carnot-Carath\\'eodory-Finsler and $d\\leq d^i \\leq \\mathbf d$.\nNotice that the sequence $(d^i)_{i \\in \\mathbb N}$ converges to a metric $d^\\infty$ due to Theorem \\ref{local maximo 1} and that we are in the conditions of Lemma \\ref{limiteintrinseco}, with $d_h = \\mathbf d$ and $d_l=d^1$.\nTherefore $d^\\infty$ is complete and intrinsic.$\\blacksquare$\n\n\\section{Existence of dense semigroups $S_X$}\n\\label{existencia sx denso}\n\nLet $G$ be a connected Lie group. \nIn this section we prove the existence of a finite subset $X=\\{x_1,\\ldots, x_k\\}\\ni e$ of $G$ such that $H_X=\\{e\\}$ and $\\bar S_X = G$.\n\nWe begin with an example.\n\n\\begin{example}\n\\label{denso na reta}\nLet $G=\\mathbb R$ be the additive group of real numbers and consider $X=\\{-1,0,\\sqrt 2\\}$.\nWe claim that $S_X$ is dense.\nLet $q \\in \\mathbb{R}$ and $\\varepsilon >0$.\nWe will find a $q_\\varepsilon \\in S_X$ such that $\\vert q - q_\\varepsilon \\vert < \\varepsilon$.\n\nIt is easy to see that there are a sequence $p_1,p_2,\\ldots$ such that $p_{i+1}-p_i = -1$ or $p_{i+1}-p_i=\\sqrt{2}$ and a infinite number of ${p_i}'s$ are in $[0,1]$.\nTherefore there exist $p_j$ and $p_k$ in $S_X$, $j<k$, such that $\\vert p_k-p_j\\vert<\\varepsilon$.\nSet $p_\\varepsilon := p_k-p_j \\in S_X$.\nIn order to fix ideas, suppose that $p_\\varepsilon >0$. \nThen there exist an element $p \\in S_X$ such that $p<q$ and a $l\\in \\mathbb N$ such that $q_\\varepsilon:=p+l.p_\\varepsilon$ satisfies $\\vert q- q_\\varepsilon \\vert < \\varepsilon$. The case $p_\\varepsilon <0$ is analogous.\nThen $S_X$ is dense in $\\mathbb R$.\nWe also have that $S_{X^{-1}}$ is dense in $G$ due to Proposition \\ref{x e menos x}.\n\nThis example is easily generalized to the case $G=\\mathbb R^n$ and $X=\\{-1,0,\\sqrt 2\\}\\times \\ldots \\times \\{-1,0,\\sqrt 2\\}$. $\\blacksquare$\n\n\\end{example}\n\nExample \\ref{denso na reta} can be generalized to Lie groups in the following way:\n\n\\begin{theorem}\n\\label{grupos de Lie}\nLet $G$ be a connected Lie group and $\\mathfrak g$ be its Lie algebra.\nLet $V$ be a vector subspace of $\\mathfrak g$ such that the Lie algebra generated by $V$ is $\\mathfrak{g}$. Let $\\{v_1,\\ldots, v_k\\}$ be a basis of $V$ such that \n\\[\n\\left\\{ \\sum_{i=1}^k a_iv_i, a_i \\in [-2,2], i=1,\\ldots, k \\right\\}\n\\]\nis contained in a neighborhood $A$ of $0 \\in \\mathfrak{g}$ such that $\\exp\\vert_A$ is a diffeomorphism over its image.\nConsider \n\\[\n\\tilde X=\\{-v_1,-v_2,\\ldots,-v_k, 0 , \\sqrt{2}v_1, \\ldots, \\sqrt{2}v_k\\}.\n\\]\nDenote $x_i=\\exp(-v_i)$, $y_i=\\exp(\\sqrt{2}v_i)$ and $X=\\{e,x_1,\\ldots,x_k,y_1,\\ldots, y_k\\}$.\nThen $\\bar S_X=G$.\n\\end{theorem} \n\n{\\it Proof}\n\n\\\n\nLet $g \\in H_X$. \nThen $gX=X$ and $e \\in X$ implies that $g \\in X$ and $g^{-1} \\in X$. \nIf $g \\neq e$, then $g=\\exp(-v_i)$ or $g=\\exp (\\sqrt{2}v_i)$.\nIn order to fix ideas, suppose that $g=\\exp(-v_i)$. \nThen we have that $g^{-1}=\\exp(v_i)\\in X$, which isn't possible because $\\exp\\vert_A$ is a diffeomorphism over its image.\nThe other case is analogous.\nConsequently $H_X=\\{e\\}$.\n\nLet us prove that $S_X$ is dense in $G$. Example \\ref{denso na reta} states that $S_{\\tilde X} \\cap \\mathbb{R}v_i$ is dense in $\\mathbb{R}v_i$ for every $i=1,\\ldots,n$.\nThis implies that $S_X \\cap \\exp(\\mathbb Rv_i)$ is dense in $\\exp(\\mathbb{R}v_i)$.\nBy Corollary \\ref{nagano para grupos de Lie}, for every $x\\in G$, there exist $m \\in \\mathbb{N}$, $t_{i_1},\\ldots, t_{i_m} \\in \\mathbb R$ and $v_{i_1},\\ldots, v_{i_m} \\in X$ such that $x=\\exp(t_{i_m}v_{i_m})\\ldots \\exp(t_{i_1}v_{i_1})$.\nBut we know that for every $j=1,\\ldots, m$, there exist a sequence of points in $S_X$ converging to $\\exp(t_{i_j}v_{i_j})$.\nTherefore we have a sequence of points in $S_X$ converging to $x$ and $S_X$ is dense in $G$.$\\blacksquare$\n\n\\section{The case $\\bar S_X=G$}\n\\label{Finlser fomula}\n\nIn this section we study the case where $\\bar S_X=G$ and $d$ is complete. \nWe prove that the sequence of induced Hausdorff metrics converges to a $C^0$-Carnot-Carath\\'eodory-Finsler metric $d^\\infty$. \nMoreover we give a necessary and sufficient condition in order to $d^\\infty$ be $C^0$-Finsler. \nIn this case, an explicit formula for the Finsler metric is obtained. \nAll these results are in Theorem \\ref{teorema principal}.\n\n\\begin{lemma}\n\\label{d barra chapeu e fracao}\nLet $G$ be a connected Lie group endowed with a metric $d$ such that $\\tau_d=\\tau_G$. Suppose that $\\hat{\\bar{d}}(x,y)= \\infty$ for some $x,y \\in G$. \nThen there exist a $v \\in \\mathfrak g$ such that \n\\[\n\\sup_{\\sigma \\in G}\\sup_{t \\neq 0} \\frac{d(\\exp(tv) \\sigma, \\sigma)}{\\vert t\\vert}\n =\\infty.\n\\]\n\\end{lemma}\n\n{\\it Proof}\n\n\\\n\nIf there exist a $\\delta > 0$ such that $\\hat{\\bar d}(\\exp(tv),e)$ is finite for every $v \\in \\mathfrak g$ and $t \\in (-\\delta,\\delta)$, then it is straightforward that $\\hat{\\bar{d}}(x,y)$ is finite for every $x,y \\in G$ due to the right invariance of $\\hat{\\bar d}$ and the connectedness of $G$. \nThen if $\\hat{\\bar{d}}(x,y)= \\infty$ for some $x,y \\in G$, then there exist $v \\in \\mathfrak g$ and $t\\in \\mathbb R$ such that $\\hat{\\bar d}(\\exp(tv),e) = \\infty$.\nIt implies that for every $C>0$, there exist a partition $\\mathcal P$ of $[0,t]$ such that\n\\[\n\\sum_{i=1}^{n_{\\mathcal P}}\\bar d(\\exp(t_iv),\\exp(t_{i-1}v)) \n= \n\\sum_{i=1}^{n_{\\mathcal P}}\\bar d(\\exp(t_i-t_{i-1})v,e) > Ct\n\\]  \ndue to the right invariance of $\\bar d$. Then the average of\n\\[\n\\frac{\\bar d(\\exp(t_i-t_{i-1})v,e)}{t_i - t_{i-1}}\n\\]\nis greater than or equal to $C$. Therefore there exist $t > 0$ such that\n\\[\n\\frac{\\bar d(\\exp(tv,e)}{t}>C.\n\\]\nThus \n\\[\n\\sup_{\\sigma \\in G}\\sup_{t \\neq 0} \\frac{d(\\exp(tv) \\sigma, \\sigma)}{\\vert t\\vert} = \\sup_{t \\neq 0}\\frac{\\bar d(\\exp(tv),e)}{t}\n =\\infty. \\blacksquare\n\\]\n\n\\begin{lemma}\n\\label{supelimsup}\nLet $G$ be a Lie group, $d$ be a metric on $G$ satisfying $\\tau_d = \\tau_G$ and $v \\in \\mathfrak g$.\nThen\n\\begin{equation}\n\\label{igualdade limsup sup 2}\n\\sup_{\\sigma \\in G}\\limsup_{t \\rightarrow 0} \\frac{d(\\exp(tv) \\sigma, \\sigma)}{\\vert t\\vert}\n =\\sup_{\\sigma \\in G}\\sup_{t \\neq 0} \\frac{d(\\exp(tv) \\sigma, \\sigma)}{\\vert t\\vert}.\n\\end{equation}\n\n\\end{lemma}\n\n{\\it Proof}\n\n\\\n\nThe inequality $\\leq $ follows directly from the definition of $\\limsup$. \nLet us prove the inequality $\\geq$.\nIf the left hand side of (\\ref{igualdade limsup sup 2}) is infinity, then there is nothing to prove.\nSuppose that it is equal to $L \\in \\mathbb R$.\nIt implies that \n\\[\n\\limsup_{t \\rightarrow 0} \\frac{d(\\exp(tv)\\sigma, \\sigma)}{\\vert t\\vert}\\leq L\n\\]\nfor every $\\sigma \\in G$.\nThen for every $\\varepsilon >0$ and $\\sigma \\in G$, there exist a $\\delta >0$ such that \n\\[\nd(\\exp(tv)\\sigma, \\sigma)\n\\leq (L + \\varepsilon)\\vert t \\vert\n\\]\nwhenever $t \\in (-\\delta, \\delta)$. \n\nNow fix $\\sigma \\in G$ and $t > 0$.\nDefine $\\gamma:[0,t] \\rightarrow G$ as $\\gamma(s) = \\exp(sv).\\sigma$.\nWe can find a partition $\\mathcal P=\\{0=t_0 < t_1 < \\ldots < t_k = t\\}$ such that \n\\begin{equation}\n\\label{limita inclinacao}\nd(\\exp(t_{i}v)\\sigma,\\exp(t_{i-1}v)\\sigma) \\leq (L + \\varepsilon)(t_i - t_{i-1}).\n\\end{equation}\nIn fact, for every $s\\in [0,t]$, we choose $\\delta_s >0 $ such that  \n\\[\nd(\\exp(\\tau v)\\sigma,\\exp(sv)\\sigma) \\leq (L + \\varepsilon)\\vert \\tau - s \\vert.\n\\] \nfor every $\\tau \\in I_s:=[0,t] \\cap (s -\\delta_s, s + \\delta_s)$. \nIf $s \\in (0,t)$, we can choose $I_s = (s -\\delta_s, s + \\delta_s)$.\nFrom $\\{I_s\\}_{s \\in [0,t]}$, we can choose a finite subcover $\\{I_{\\tilde k}\\}$. \nFrom this finite subcover, we drop a $I_{\\tilde k_1}$, once a time, if there exist $I_{\\tilde k_2}$ such that $I_{\\tilde k_1} \\subset I_{\\tilde k_2}$.\nWe end up with a family such that one element is not contained in another element. \nDenote this family by $\\{[0,0 + \\delta_0),(t_2 - \\delta_{t_2},t_2 + \\delta_{t_2}),(t_4 - \\delta_{t_4}, t_4 + \\delta_{t_4}), \\ldots,(t-\\delta_t,t]\\}$, with $t_{i-2} < t_i$ for every $i$ ($t_{i-2} = t_i$ doesn't happen).\nIt is not difficult to see that $0 < t_2-\\delta_{t_2} < t_4 - \\delta_{t_4} < \\ldots < t - \\delta_t$  and $0 + \\delta_0 < t_2 + \\delta_{t_2} < t_4 + \\delta_{t_4} < \\ldots < t$. \nNow we choose the ``odd'' points. \n$t_1 \\in (0,t_2)$ is chosen in $[0,0 + \\delta_0) \\cap (t_2 - \\delta_{t_2},t_2 + \\delta_{t_2})$. \n$t_3 \\in (t_2,t_4)$ is chosen in $ (t_2 - \\delta_{t_2},t_2 + \\delta_{t_2}) \\cap (t_4 - \\delta_{t_4}, t_4 + \\delta_{t_4}) $ and so on.\nThen $\\mathcal P=\\{0=t_0 < t_1 < \\ldots < t_k = t\\}$ is such that (\\ref{limita inclinacao}) is satisfied.\n\nTherefore \n\\[\n\\begin{array}{ccl}\nd(\\exp(tv)\\sigma,\\sigma) & \\leq & \\sum\\limits_{i=1}^{n_\\mathcal P} d(\\exp(t_{i}v)\\sigma,\\exp(t_{i-1}v)\\sigma) \\leq (L+\\varepsilon)\\vert t\\vert\n\\end{array}\n\\]\nfor every $\\sigma$ and $t>0$ and we can conclude that\n\\[\n\\sup_{\\sigma \\in G}\\sup_{t > 0} \\frac{d(\\exp(tv) \\sigma, \\sigma)}{\\vert t\\vert} \\leq L.\n\\]\n\nIf $t<0$, then\n\\begin{eqnarray}\n\\sup_{\\sigma \\in G}\\sup_{t < 0} \\frac{d(\\exp(tv) \\sigma, \\sigma)}{\\vert t\\vert} & = & \\sup_{\\sigma \\in G}\\sup_{t > 0} \\frac{d(\\exp(t(-v)) \\sigma, \\sigma)}{\\vert t\\vert} \\nonumber \\\\ \n& \\leq & \\sup_{\\sigma \\in G}\\limsup_{t \\rightarrow 0} \\frac{d(\\exp(t(-v)) \\sigma, \\sigma)}{\\vert t\\vert} \\nonumber \\\\\n& = & \\sup_{\\sigma \\in G}\\limsup_{t \\rightarrow 0} \\frac{d(\\exp(tv) \\sigma, \\sigma)}{\\vert t\\vert} = L,\n\\end{eqnarray}\nwhat settles the proposition.$\\blacksquare$\n\n\\\n\nIn what follows we identify $G \\times \\mathfrak g$ with $TG$ through the correspondence $(g,v) \\mapsto (g,dR_g(v))$, where $R_g$ denotes the right translation by $g$ on $G$.\n\n\\begin{theorem}\n\\label{teorema principal}\nLet $G$ be a connected Lie group endowed with a complete metric $d$ such that $\\tau_d=\\tau_G$, $\\varphi: G \\times G \\rightarrow G$ be the product of $G$ and $X=\\{x_1,\\ldots x_k\\} \\ni e$ be a finite subset of $G$ such that $H_X=\\{e\\}$ and $\\bar S_X=G$.\n\\begin{enumerate}\n\\item If $d$ is bounded above by a right invariant intrinsic metric $\\mathbf d$, then the sequence of induced Hausdorff metrics converges to a right invariant $C^0$-Carnot-Carath\\'eodory-Finsler metric $d^\\infty$.\nIn particular, if $\\hat{\\bar{d}}$ is a metric, then $d^\\infty$ is a right invariant $C^0$-Carnot-Carath\\'eodory-Finsler.\n\\item If $d$ is bounded above by a right invariant $C^0$-Finsler metric $\\mathbf d$, then the sequence of induced Hausdorff metrics converges to a right invariant $C^0$-Finsler metric $d^\\infty$.\n\\item Suppose that the sequence of induced Hausdorff metrics converges pointwise to a metric $d^\\infty$.\nThen $d^\\infty$ is $C^0$-Finsler iff \n\\begin{equation}\n\\label{f tilde}\n\\tilde F(g,v):=\\sup_{\\sigma\\in G}\\limsup_{t \\rightarrow 0}\\frac{d(\\exp(tv)g\\sigma,g\\sigma)}{\\vert t \\vert}\n\\end{equation}\nis finite for every $(g,v) \\in TG$, and in this case $\\tilde F$ is the Finsler metric on $G$.\n\\end{enumerate}\n\\end{theorem}\n\n{\\it Proof}\n\n\\\n\n(1) We know that the sequence of induced Hausdorff metrics converges to a intrinsic metric $d^\\infty$ due to Theorem \\ref{completo eh intrinseco}. \nIt is a right invariant due to Lemma \\ref{translacao a direita decresce}.\nTherefore $d^\\infty$ is $C^0$-Carnot-Carath\\'eodory-Finsler due to Theorem \\ref{Berestovskii theorem}.\n\nIf $\\hat{\\bar{d}}$ is finite, then $\\mathbf d=\\hat{\\bar{d}}$ is a right invariant intrinsic bound for $d$ and the result follows.\n\n\\\n\n(2) This is a particular case of item (1) and it is direct consequence of the second paragraph of Remark \\ref{relacoes Carnot Caratheodory}.\n\n\\\n\n(3) First of all we prove that if $\\tilde F$ is finite, then $d^\\infty$ is $C^0$-Finsler.\n\nIf $\\tilde F$ is finite, then $\\hat{\\bar d}$ is also finite due to Lemmas \\ref{d barra chapeu e fracao} and \\ref{supelimsup}.\nThus $d^\\infty$ is a right invariant $C^0$-Carnot-Carath\\'eodory-Finsler due to item (1).\n\nIt is a direct consequence of the ball-box theorem that if $d^\\infty$ is a Carnot-Carath\\'eodory metric on a differentiable manifold $M$ with respect to a proper smooth distribution $\\mathcal D$, then for every $p \\in M$ and $v \\not\\in \\mathcal D_p$ we have that\n\\[\n\\lim_{t \\rightarrow 0}\\frac{d^\\infty (\\gamma(t), p)}{\\vert t \\vert}=\\infty,\n\\]\nwhere $\\gamma:(-\\varepsilon,\\varepsilon) \\rightarrow M$ is a smooth path such that $\\gamma(0)=p$ and $\\gamma^\\prime(0)=v$.\nThe same conclusion holds for $C^0$-Carnot-Carath\\'eodory-Finsler metrics due to Remark \\ref{relacoes Carnot Caratheodory}.\nThen $d^\\infty$ is $C^0$-Finsler whenever $\\tilde F$ is finite.\n\nNow we prove that if $d^\\infty$ is $C^0$-Finsler, then $\\tilde F$ is finite.\n\nIf $F:TG \\rightarrow \\mathbb R$ is the $C^0$-Finsler metric correspondent to $d^\\infty$, then\n\\[\nF(g,v)\n=\\lim_{t\\rightarrow 0} \\frac{d^\\infty(\\exp(tv)g,g)}{\\vert t\\vert} \n= \\sup_{\\sigma \\in G}\\lim_{t\\rightarrow 0} \\frac{d^\\infty(\\exp(tv)g \\sigma,g \\sigma)}{\\vert t\\vert} \\geq \\tilde F(g,v),\n\\]\nwhere the first equality is due to Theorem 3.7 of \\cite{BenettiFukuoka} and the second equality is due to the right invariance of $d^\\infty$.\nTherefore $\\tilde F$ is finite.\nObserve that this part of the proof also shows that $\\tilde F \\leq F$ if $d^\\infty$ is $C^0$-Finsler.\n\nFinally we show that if $d^\\infty$ is $C^0$-Finsler, then $F \\leq \\tilde F$.\n\nFix $t>0$ and $v \\in \\mathfrak g$. Define $\\gamma:[0,t] \\rightarrow G$ by $\\gamma(s)=\\exp(sv)$ and notice that if $\\xi \\in G$, then\n\n\\begin{eqnarray}\nd^1(\\exp(tv)\\xi, \\xi) \n& \n\\leq \n&\n\\ell_{d_X} (\\gamma \\xi)\n=  \n\\sup\\limits_{\\mathcal P} \\sum\\limits_{i=1}^{n_{\\mathcal P}} d_X(\\exp(t_i v)\\xi,\\exp(t_{i-1}v)\\xi) \\nonumber \\\\\n& \n= \n& \\sup\\limits_{\\mathcal P} \\sum\\limits_{i=1}^{n_{\\mathcal P}} \\max\\limits_j d(\\exp(t_i v)\\xi x_j,\\exp(t_{i-1}v)\\xi  x_j) \\nonumber \\\\\n&\n= \n& \\sup\\limits_{\\mathcal P} \\sum\\limits_{i=1}^{n_{\\mathcal P}} \\max\\limits_j d(\\exp((t_i-t_{i-1}) v)h_j,h_j) \\nonumber \\\\\n&\n\\leq\n& \\sup\\limits_{\\mathcal P} \\sum\\limits_{i=1}^{n_{\\mathcal P}} \\sup\\limits_{\\sigma \\in G} d(\\exp((t_i-t_{i-1}) v) \\sigma, \\sigma) \\nonumber\n\\end{eqnarray}\n\nwhere $h_j = \\exp(t_{i-1}v)\\xi x_j$. Then\n\\[\n\\begin{array}{ccl}\nd^1(\\exp(tv)\\xi, \\xi) \n& \n\\leq \n&\n\\sup\\limits_{\\mathcal P} \\sum\\limits_{i=1}^{n_{\\mathcal P}} \\tilde F(v)(t_i-t_{i-1}) = \\tilde F(v)t,\n\\end{array}\n\\]\nwhere the inequality is due to Lemma \\ref{supelimsup}. \nIf we iterate this process, replacing $d^1$ by $d^2, d^3, \\ldots$, we get \n\\begin{equation}\n\\label{compara com is}\nd^i(\\exp(tv)\\xi,\\xi) \\leq \\tilde F(v) t\n\\end{equation} \nfor every $i \\in \\mathbb N$, $t > 0$ and $\\xi \\in G$. \nBut $\\tilde F(v)=\\tilde F(-v)$ because\n\\begin{eqnarray}\n\\tilde F(v)\n&\n=\n&\n\\sup_{\\sigma\\in G}\\limsup_{t \\rightarrow 0}\\frac{d(\\exp(tv)\\sigma,\\sigma)}{\\vert t \\vert} \\nonumber \\\\\n&\n=\n&\n\\sup_{\\sigma\\in G}\\limsup_{t \\rightarrow 0}\\frac{d(\\exp(tv)\\exp(-tv)\\sigma,\\exp(-tv)\\sigma)}{\\vert t \\vert} \n= \\tilde F(-v),\n\\end{eqnarray}\nand\n\\begin{equation}\n\\label{d infinito e f til}\nd^\\infty(\\exp(tv)\\sigma,\\sigma) \\leq \\tilde F(v)\\vert t \\vert\n\\end{equation}\nholds for every $t \\in \\mathbb R$ and $\\sigma \\in G$ due to (\\ref{compara com is}). Therefore\n\\[\nF(v)\n=\n\\lim_{t \\rightarrow 0}\\frac{d^\\infty(\\exp(tv),e)}{\\vert t \\vert}\\leq \\tilde F(v),\n\\]\nwhat settles the theorem.$\\blacksquare$\n\n\\section{Further examples}\n\\label{secao exemplos}\n\n\\begin{example}\nConsider the additive group $G=\\mathbb R$ with the finite metric $d(x,y)=\\vert\\arctan (x) - \\arctan (y) \\vert$.\nLet $X=\\{ -1,0,\\sqrt 2\\}$.\nThen $\\bar S_X=\\mathbb R$ (see Example \\ref{denso na reta}).\nIts maximum derivative is equal to one and it is not difficult to see that $\\hat{\\bar d}$ is a metric on $G$.\nThen $d^\\infty$ is an invariant $C^0$-Finsler metric, with $F(\\cdot)$ equal to the Euclidean norm (See Theorem \\ref{teorema principal}).\nTherefore $d^\\infty$ is the Euclidean metric.\n\nNow we prove that every $d^i$ of the sequence of induced Hausdorff metrics is a finite metric.\nFirst of all notice that\n\\[\n\\frac{d}{dt}\\arctan t = \\frac{1}{1+t^2}.\n\\]\nIt means that if $0<x<y$, then \n\\[\nd(x,y) = \\arctan y - \\arctan x \\leq \\frac{1}{1+x^2} (y - x).\n\\]\nbecause $t \\mapsto \\arctan t$ has the concavity pointed downwards for $t >0$.\nFor $1<x<y$, we have that\n\\[\nd^1(x,y)=\\sup_{\\mathcal P}\\sum_{i=1}^{n_{\\mathcal P}} \\max_j d(x+x_j,y+x_j) \\leq \\frac{1}{1+(x-1)^2}(y-x).\n\\]\nWe can iterate this procedure and show that if $i \\in \\mathbb N$ and $i < x< y$, then \n\\[\nd^i(x,y) \\leq \\frac{1}{1+(x-i)^2}(y-x).\n\\]\nTherefore, for a fixed $i \\in \\mathbb N$, we have that\n\\[\n\\lim_{x\\rightarrow \\infty} d^i(i+1,x) \\leq \\sum_{j=1}^\\infty d^i(i+j,i+j+1) \\leq \\sum_{j=1}^\\infty \\frac{1}{1+j^2} < \\infty.\n\\]\nThe finiteness of the negative part of $\\mathbb R$ follows analogously.\nTherefore $d^i$ is a finite metric (and therefore it isn't complete) for every $i \\in \\mathbb N$ despite $d^\\infty$ is complete, what shows that the completeness of $d$ is not necessary to make $d^\\infty$ intrinsic.\n\n\\end{example}\n\n\\begin{example}\n\\label{exemplo diverge}\nConsider the additive group $\\mathbb R$  endowed with the metric $d(x,y)=\\vert \\sqrt[3]{x} - \\sqrt[3]{y} \\vert$ and let $X=\\{-1,0,\\sqrt{2}\\}$.\nSuppose that the sequence of induced Hausdorff metric converges to a metric.\n$\\mathbb R$ doesn't admit a $C^0$-Carnot-Carath\\'eodory-Finsler metric which isn't a $C^0$-Finsler metric, because the only distribution that can generate a $C^0$-Carnot-Carath\\'eodory-Finsler metric is the whole tangent bundle.\nTherefore the sequence converges to a $C^0$-Finsler metric.\nBut in this case it is straightforward to see that $\\tilde F$ in (\\ref{f tilde}) is infinite (just analyze small neighborhoods of the origin), what contradicts Theorem \\ref{teorema principal}.\nTherefore the sequence of induced Hausdorff metrics doesn't converge to a metric on $\\mathbb R$.\n\\end{example}\n\n\\begin{example}\n\\label{flat torus}\nConsider the flat torus $T=S^1 \\times S^1$ with the canonical group operation.\nWe represent it as $[0,1]\\times [0,1]$ endowed with the canonical metric with the opposite sides identified and endowed with the quotient metric $d$.\nConsider $X=T-Q$, where $Q$ is the equivalence class of the square $(1\/4,3\/4) \\times (1\/4,3\/4) \\subset [0,1] \\times [0,1]$ in $T^2$. \nIt is not difficult to see that $d_X$ is the maximum metric when restricted to a small neighborhood of the identity element. \nTherefore $d^1$ is locally equal to the maximum metric. \n(Compare with Example 8.11 in \\cite{BenettiFukuoka}).\nIn this case we have that $d^1 < d$, a relationship that doesn't happen in this work.\nThis example shows that the sequence of induced Hausdorff metrics are not only about increasing sequences of metrics.\n\\end{example}\n\n\\section{Final remarks}\n\\label{secao final}\n\n\\begin{remark}\nWe don't know if Theorem \\ref{teorema principal} holds if $d$ isn't complete.\nMore in general, we don't know if the limit of an increasing sequence of intrinsic metrics is intrinsic.\n\\end{remark}\n\n\\begin{remark}\nFrom a Lie group $G$ endowed with a metric satisfying mild conditions, we defined a discrete dynamical system $d,d^1, \\ldots$ that converges to a right invariant $C^0$-Carnot-Carath\\'eodory-Finsler metric.\nThis is the first example that shows that a sequence of metrics defined from the Hausdorff distance can have a regularizing effect.\nMoreover, for a general compact subset $X \\subset G$, the sequence of induced Hausdorff metrics is not necessarily increasing, as Example \\ref{flat torus} shows, what makes the dynamics more interesting.\nDue to these facts we think that it is worthwhile to study the properties of the sequence of induced Hausdorff metrics in other situations.\n\\end{remark}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\nLanguage model plays an important role in many natural language processing systems, such as in automatic speech recognition \\cite{rabiner1993fundamentals,chan2016listen} and machine translation systems \\cite{brown1990statistical,cho2014learning}. Recurrent neural network (RNN) based models \\cite{mikolov2010recurrent,mikolov2011extensions} have recently shown success in language modeling, outperforming conventional n-gram based models. Long short-term memory \\cite{hochreiter1997long,sundermeyer2012lstm} is a widely used RNN variant for language modeling due to its superior performance in capturing longer term dependencies.\n\nConventional RNN based language model uses a hidden state to represent the summary of the preceding words in a sentence without considering context signals. Mikolov et al. proposed a context dependent RNN language model \\cite{mikolov2012context} by connecting a contextual vector to the RNN hidden state. This contextual vector is produced by applying Latent Dirichlet Allocation \\cite{blei2003latent} on preceding text. Several other contextual language models were later proposed by using bag-of-word \\cite{wang2015larger} and RNN methods \\cite{ji2015document} to learn larger context representation that beyond the target sentence. \n\nThe previously proposed contextual language models treat preceding sentences as a sequence of inputs, and they are suitable for document level context modeling. In dialog modeling, however, dialog interactions between speakers play an important role. Modeling utterances in a dialog as a sequence of inputs might not well capture the pauses, turn-taking, and grounding phenomena \\cite{clark1991grounding} in a dialog. In this work, we propose contextual RNN language models that specially track the interactions between speakers. We expect such models to generate better representations of the dialog context.\n\nThe remainder of the paper is organized as follows. In section 2, we introduce the background on contextual language modeling. In section 3, we describe the proposed dialog context language models. Section 4 discusses the evaluation procedures and results. Section 5 concludes the work.\n\n\\section{Background}\n\\label{sec:background}\n\n\\subsection{RNN Language Model}\nA language model assigns a probability to a sequence of words $\\mathbf{w}=(w_1, w_2, ..., w_{T})$ following probability distribution. \nUsing the chain rule, the likelihood of the word sequence $\\mathbf{w}$ can be factorized as:\n        \\begin{equation}\n            P(\\mathbf{w}) = P(w_1, w_2, ..., w_{T}) = \\prod_{t=1}^{T}P(w_{t}|w_{< t}) \\\\\n        \\end{equation}\nAt time step $t$, the system input is the embedding of the word at index $t$, and the system output is the probability distribution of the word at index $t+1$. The RNN hidden state $h_t$ encodes the information of the word sequence up till current step:\n        \\begin{align}\n            &h_t = \\operatorname{RNN}(h_{t-1}, w_t) \\\\\n            &P(w_{t+1}|w_{< t+1}) = \\operatorname{softmax}(W_{o}h_{t} + b_{o})\n        \\end{align}\nwhere $W_{o}$ and $b_{o}$ are the output layer weights and biases.\n\n\\subsection{Contextual RNN Language Model}\nA number of methods have been proposed to introduce contextual information to the language model. Mikolov and Zweig \\cite{mikolov2012context} proposed a topic-conditioned RNNLM by introducing a contextual real-valued vector to RNN hidden state. The contextual vector was created by performing LDA \\cite{blei2003latent} on preceding text. Wang and Cho \\cite{wang2015larger} studied introducing corpus-level discourse information into language modeling. A number of context representation methods were explored, including bag-of-words, sequence of bag-of-words, and sequence of bag-of-words with attention. Lin et al. \\cite{lin2015hierarchical} proposed using hierarchical RNN for document modeling. Comparing to using bag-of-words and sequence of bag-of-words for document context representation, using hierarchical RNN can better model the order of words in preceding text, at the cost of the increased computational complexity. These contextual language models focused on contextual information at the document level. Tran et al. \\cite{tran2016inter} further proposed a contextual language model that consider information at inter-document level. They claimed that by utilizing the structural information from a tree-structured document set, language modeling performance was largely improved. \n\n\\section{Methods}\n\\label{sec:methods}\n\nThe previously proposed contextual language models focus on applying context by encoding preceding text, without considering interactions in dialogs. These models may not be well suited for dialog language modeling, as they are not designed to capture dialog interactions, such as clarifications and confirmations. By making special design in learning dialog interactions, we expect the models to generate better representations of the dialog context, and thus lower perplexity of the target dialog turn or utterance. \n\nIn this section, we first explain the context dependent RNN language model that operates on utterance or turn level. Following that, we describe the two proposed contextual language models that utilize the dialog level context.\n\n\\subsection{Context Dependent RNNLM}\nLet $\\mathbf{D} = (\\mathbf{U}_1, \\mathbf{U}_2, ..., \\mathbf{U}_K)$ be a dialog that has $K$ turns and involves two speakers. Each turn may have one or more utterances. The $k$th turn $\\mathbf{U}_k = (w_1, w_2, ..., w_{T_k})$ is represented as a sequence of $T_k$ words. \nConditioning on information of the preceding text in the dialog, probability of the target turn $\\mathbf{U}_k$ can be calculated as:\n        \\begin{equation}\n            P(\\mathbf{U}_k|\\mathbf{U}_{<k}) = \\prod_{t=1}^{T_k}P(w^{\\mathbf{U}_{k}}_{t}|w^{\\mathbf{U}_{k}}_{< t}, \\mathbf{U}_{<k}) \\\\\n        \\end{equation}\nwhere $\\mathbf{U}_{<k}$ denotes all previous turns before $\\mathbf{U}_k$, and $w^{\\mathbf{U}_{k}}_{< t}$ denotes all previous words before the $t$th word in turn $\\mathbf{U}_k$.\n\n        \\begin{figure}[t]\n            \\centering\n            \\includegraphics[width=200pt]{Context_RNNLM_1.pdf}\n            \\caption{{ Context dependent RNN language model.}}\n            \\label{fig:Context_RNNLM}\n        \\end{figure}\nIn context dependent RNN language model, the context vector $c$ is connected to the RNN hidden state together with the input word embedding at each time step (Figure \\ref{fig:Context_RNNLM}). This is similar to the context dependent RNN language model proposed in \\cite{mikolov2012context}, other than that the context vector is not connected directly to the RNN output layer. With the additional context vector input $c$, the RNN state $h_t$ is updated as:\n        \\begin{equation}\n            h_t = \\operatorname{RNN}(h_{t-1}, [w_t, c])\n        \\end{equation}\n\n\\subsection{Context Representations}\nIn neural network based language models, the dialog context can be represented as a dense continuous vector. This context vector can be produced in a number of ways. \n\n\\begin{comment}\n    All the words in the preceding text can be bagged into a single vector $l = [0,1]^{|v|}$, with $|v|$ being the vocabulary size, and each element being the word frequency in the text. With the word embedding matrix $W \\in \\mathbb{R}^{d \\times |v|}$, where $d$ denotes embedding size, the bag of word embedding context vector can be calculated as $c = Wl$.\n\\end{comment}\nOne simple approach is to use bag of word embeddings. However, bag of word embedding context representation does not take word order into consideration. An alternative approach is to use an RNN to read the preceding text. The last hidden state of the RNN encoder can be seen as the representation of the text and be used as the context vector for the next turn. To generate document level context representation, one may cascade all sentences in a document by removing the sentence boundaries. The last RNN hidden state of the previous utterance serves as the initial RNN state of the next utterance. As in \\cite{ji2015document}, we refer to this model as DRNNLM. Alternatively, in the CCDCLM model proposed in \\cite{ji2015document}, the last RNN hidden state of the previous utterance is fed to the RNN hidden state of the target utterance at each time step.\n\n\\subsection{Interactive Dialog Context LM}\nThe previously proposed contextual language models, such as DRNNLM and CCDCLM, treat dialog history as a sequence of inputs, without modeling dialog interactions. A dialog turn from one speaker may not only be a direct response to the other speaker's query, but also likely to be a continuation of his own previous statement. Thus, when modeling turn $k$ in a dialog, we propose to connect the last RNN state of turn $k-2$ directly to the starting RNN state of turn $k$, instead of letting it to propagate through the RNN for turn $k-1$. The last RNN state of turn $k-1$ serves as the context vector to turn $k$, which is fed to turn $k$'s RNN hidden state at each time step together with the word input. The model architecture is as shown in Figure \\ref{fig:IDCLM}. The context vector $c$ and the initial RNN hidden state for the $k$th turn $h^{\\mathbf{U}_k}_{0}$ are defined as:\n        \\begin{equation}\n            c = h^{\\mathbf{U}_{k-1}}_{T_{k-1}}, \\; h^{\\mathbf{U}_k}_{0} = h^{\\mathbf{U}_{k-2}}_{T_{k-2}}\n        \\end{equation}\nwhere $h^{\\mathbf{U}_{k-1}}_{T_{k-1}}$ represents the last RNN hidden state of turn $k-1$. This model also allows the context signal from previous turns to propagate through the network in fewer steps, which helps to reduce information loss along the propagation. We refer to this model as Interactive Dialog Context Language Model (IDCLM).\n        \\begin{figure}[h]\n            \\centering\n            \\includegraphics[width=240pt]{ICDCLM.pdf}\n            \\caption{{Interactive Dialog Context Language Model (IDCLM).}}\n            \\label{fig:IDCLM}\n        \\end{figure}\n\\subsection{External State Interactive Dialog Context LM}\nThe propagation of dialog context can be seen as a series of updates of a hidden dialog context state along the growing dialog. IDCLM models this hidden dialog context state changes implicitly in the turn level RNN state. Such dialog context state updates can also be modeled in a separated RNN. As shown in the architecture in Figure \\ref{fig:ESIDCLM}, we use an external RNN to model the context changes explicitly. Input to the external state RNN is the vector representation of the previous dialog turns. The external state RNN output serves as the dialog context for next turn:\n        \\begin{equation}\n           \n            s_{k-1} = \\operatorname{RNN}_{ES}(s_{k-2}, h^{\\mathbf{U}_{k-1}}_{T_{k-1}})\n        \\end{equation}\nwhere $s_{k-1}$ is the output of the external state RNN after the processing of turn $k-1$. The context vector $c$ and the initial RNN hidden state for the $k$th turn $h^{\\mathbf{U}_k}_{0}$ are then defined as:\n        \\begin{equation}\n            c = s_{k-1}, \\; h^{\\mathbf{U}_k}_{0} = h^{\\mathbf{U}_{k-2}}_{T_{k-2}}\n        \\end{equation}\nWe refer to this model as External State Interactive Dialog Context Language Model (ESIDCLM). \n        \\begin{figure}[h]\n            \\centering\n            \\includegraphics[width=240pt]{EICDCLM_1.pdf}\n            \\caption{{ External State Interactive Dialog Context Language Model (ESIDCLM). }}\n            \\label{fig:ESIDCLM}\n        \\end{figure}\n        \nComparing to IDCLM, ESIDCLM releases the burden of turn level RNN by using an external RNN to model dialog context state changes. One drawback of ESIDCLM is that there are additional RNN model parameters to be learned during model training, which may make the model more prone to overfitting when training data size is limited.\n\n\n\\section{Experiments}\n\\label{sec:experiments}\n\n\\subsection{Data Set}\nWe use the Switchboard Dialog Act Corpus (SwDA)\\footnote{http:\/\/compprag.christopherpotts.net\/swda.html} in evaluating our contextual langauge models. The SwDA corpus extends the Switchboard-1 Telephone Speech Corpus with turn and utterance-level dialog act tags. The utterances are also tagged with part-of-speech (POS) tags. We split the data in folder sw00 to sw09 as training set, folder sw10 as test set, and folder sw11 to sw13 as validation set. The training, validation, and test sets contain 98.7K turns (190.0K utterances), 5.7K turns (11.3K utterances), and 11.9K turns (22.2K utterances) respectively. Maximum turn length is set to 160. The vocabulary is defined with the top frequent 10K words. \n\n\\subsection{Baselines}\nWe compare IDCLM and ESIDCLM to several baseline methods, including n-gram based model, single turn RNNLM, and various context dependent RNNLMs. \n\n\\textbf{5-gram KN} \\hspace{3mm} A 5-gram language model with modified Kneser-Ney smoothing \\cite{chen1996empirical}.\n\n\\textbf{Single-Turn-RNNLM} \\hspace{3mm} Conventional RNNLM that operates on single turn level with no context information.\n\n\\textbf{BoW-Context-RNNLM} \\hspace{3mm} Contextual RNNLM with BoW representation of preceding text as context.\n\n\\textbf{DRNNLM} \\hspace{3mm} Contextual RNNLM with turn level context vector connected to initial RNN state of the target turn.\n\n\\textbf{CCDCLM} \\hspace{3mm} Contextual RNNLM with turn level context vector connected to RNN hidden state of the target turn at each time step. We implement this model following the design in \\cite{ji2015document}.\n\nIn order to investigate the potential performance gain that can be achieved by introducing context, we also compare the proposed methods to RNNLMs that use true dialog act tags as context. Although human labeled dialog act might not be the best option for modeling the dialog context state, it provides a reasonable estimation of the best gain that can be achieved by introducing linguistic context. The dialog act sequence is modeled by a separated RNN, similar to the external state RNN used in ESIDCLM. We refer to this model as Dialog Act Context Language Model (DACLM). \n\n\\textbf{DACLM} \\hspace{3mm} RNNLM with true dialog act context vector connected to RNN state of the target turn at each time step.\n\n\\subsection{Model Configuration and Training}\nIn this work, we use LSTM cell \\cite{hochreiter1997long} as the basic RNN unit for its stronger capability in capturing long-range dependencies in a word sequence comparing to simple RNN. We use pre-trained word vectors~\\cite{mikolov2013distributed} that are trained on Google News dataset to initialize the word embeddings. These word embeddings are fine-tuned during model training. We conduct mini-batch training using Adam optimization method following the suggested parameter setup in~\\cite{kingma2014adam}. Maximum norm is set to 5 for gradient clipping . For regularization, we apply dropout ($p=0.8$) on the non-recurrent connections~\\cite{zaremba2014recurrent} of LSTM. In addition, we apply $L_2$ regularization ($\\lambda = 10^{-4}$) on the weights and biases of the RNN output layer.\n\n\\subsection{Results and Analysis}\nThe experiment results on language modeling perplexity for models using different dialog turn size are shown in Table \\ref{table:results}. $K$ value indicates the number of turns in the dialog. Perplexity is calculated on the last turn, with preceding turns used as context to the model. \n\n        \\begin{table} [h]\n        \\caption{\\label{table1} {Language modeling perplexities on SwDA corpus with various dialog context turn sizes (K).}}\n        \\vspace{2mm}\n        \\centerline{\n        \\begin{tabular}{l c c c c}\n        \\hline\n        \\textbf{Model} & \\textbf{K=1} & \\textbf{K=2}  & \\textbf{K=3}  & \\textbf{K=5}\\\\\n        \\hline \n        5-gram KN & 65.7 & - & - & -  \\\\\n        Single-Turn-RNNLM & 60.4 & - & - & - \\\\\n        BoW-Context-RNNLM & -    & 59.6 & 59.2 & 58.9   \\\\\n        DRNNLM            & -    & 60.1 & 58.6 & 59.1 \\\\ \n        CCDCLM            & -    & 63.9 & 61.4 & 62.2 \\\\ \n        \\hline\n        IDCLM             & -    & - & 58.8 & 58.6  \\\\ \n        ESIDCLM           & -    & - & \\textbf{58.4} & \\textbf{58.5}  \\\\ \n        \\hline\n        DACLM  & -    & 58.2 & 57.9 & 58.0  \\\\ \n        \\hline\n        \\end{tabular}\n        \\label{table:results}\n        }\n        \\end{table}\n\nAs can be seen from the results, all RNN based models outperform the n-gram model by large margin. The BoW-Context-RNNLM and DRNNLM beat the Single-Turn-RNNLM consistently. Our implementation of the context dependent CCDCLM performs worse than Single-Turn-RNNLM. This might due to fact that the target turn word prediction depends too much on the previous turn context vector, which connects directly to the hidden state of current turn RNN at each time step. The model performance on training set might not generalize well during inference given the limited size of the training set. \n\nThe proposed IDCLM and ESIDCLM beat the single turn RNNLM consistently under different context turn sizes. ESIDCLM shows the best language modeling performance under dialog turn size of 3 and 5, outperforming IDCLM by a small margin. IDCLM beats all baseline models when using dialog turn size of 5, and produces slightly worse perplexity than DRNNLM when using dialog turn size of 3.\n\nTo analyze the best potential gain that may be achieved by introducing linguistic context, we compare the proposed contextual models to DACLM, the model that uses true dialog act history for dialog context modeling. As shown in Table \\ref{table:results}, the gap between our proposed models and DACLM is not wide. This gives a positive hint that the proposed contextual models may implicitly capture the dialog context state changes. \n\nFor fine-grained analyses of the model performance, we further compute the test set perplexity per POS tag and per dialog act tag. We selected the most frequent POS tags and dialog act tags in SwDA corpus, and report the tag based perplexity relative changes ($\\%$) of the proposed models comparing to Single-Turn-RNNLM. A negative number indicates performance gain.\n\n        \\begin{table} [h]\n        \\caption{ {Perplexity relative change ($\\%$) per POS tag }}\n        \\vspace{2mm}\n        \\centerline{\n        \\begin{tabular}{l c c c}\n        \\hline\n        \\textbf{POS Tag} & \\textbf{IDCLM} & \\textbf{ESIDCLM} & \\textbf{DACLM}\\\\\n        \\hline \n        PRP & -16.8 & -5.8 & -10.1 \\\\ \n        IN & -2.0 & -5.5 & -1.8 \\\\ \n        RB & -4.1 & -8.9 & -4.3 \\\\ \n        NN & 13.4 & 8.1 & 2.3 \\\\ \n        UH & -0.4 & 7.7 & -9.7 \\\\ \n        \\hline\n        \\end{tabular}\n        \\label{table:pos_tag_results}\n        }\n        \\end{table}\n\nTable \\ref{table:pos_tag_results} shows the model perplexity per POS tag. All the three context dependent models produce consistent performance gain over the Single-Turn-RNNLM for pronouns, prepositions, and adverbs, with pronouns having the largest perplexity improvement. However, the proposed contextual models are less effective in capturing nouns. This suggests that the proposed contextual RNN language models exploit the context to achieve superior prediction on certain but not all POS types. Further exploration on the model design is required if we want to better capture words of a specific type.\n\n        \\begin{table} [h]\n        \\caption{ {Perplexity relative change ($\\%$) per dialog act tag. }}\n        \\vspace{2mm}\n        \\centerline{\n        \\begin{tabular}{l c c c}\n        \\hline\n        \\textbf{DA Tag} & \\textbf{IDCLM} & \\textbf{ESIDCLM} & \\textbf{DACLM}\\\\\n        \\hline \n        Statement-non-opinion & -1.8 & -0.5 & -1.6 \\\\ \n        Acknowledge & -2.6 & 11.4 & -16.3 \\\\ \n        Statement-opinion & 4.9 & -0.9 & -1.0 \\\\ \n        Agree\/Accept & 14.7 & 2.7 & -15.1 \\\\ \n        Appreciation & 0.7 & -3.8 & -6.5 \\\\\n        \\hline\n        \\end{tabular}\n        \\label{table:da_tag_results}\n        }\n        \\end{table}\n        \nFor the dialog act tag based results in Table \\ref{table:da_tag_results}, the three contextual models show consistent performance gain on Statement-non-opinion type utterances. The perplexity changes for other dialog act tags vary for different models. \n\n\n\\section{Conclusions}\n\\label{sec:conclusions}\nIn this work, we propose two dialog context language models that with special design to model dialog interactions. Our evaluation results on Switchboard Dialog Act Corpus show that the proposed model outperform conventional RNN language model by 3.3\\%. The proposed models also illustrate advantageous performance over several competitive contextual language models. Perplexity of the proposed dialog context language models is higher than that of the model using true dialog act tags as context by a small margin. This indicates that the proposed model may implicitly capture the dialog context state for language modeling. \n\n\\bibliographystyle{IEEEbib}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nCurrent atmospheric and accelerator-based experiments study low energy\nneutrino interactions to optimize their sensitivity to neutrino oscillations.\nMaximum sensitivity requires precise knowledge of the cross sections for\ncharged current (CC) and neutral current (NC) neutrino-nucleus interactions\nin the sub-to-few GeV range. The most relevant inputs include\nquasi-elastic (QE) scattering cross sections and kinematics for accurate \nprediction of signal rates in these experiments. In addition, reliable\nNC $\\pi^0$ production cross sections are needed for estimation of the \ndominant backgrounds for $\\numu \\rightarrow \\nue$ searches, CC $\\pi^+$ \nproduction rates for estimation of the dominant \nbackgrounds for $\\numu$ disappearance searches, and antineutrino cross sections\nfor both signal and background predictions in CP violation searches. \nBecause intense accelerator-based low energy neutrino beams have only \nbeen in operation for a few years, the information on these low \nenergy neutrino cross sections comes primarily from bubble chamber, spark \nchamber, and emulsion experiments that ran decades ago. While \nextremely valuable for constraining present neutrino simulations, many of \nthese experimental results suffered from low statistics and large \nneutrino flux uncertainties. As a result, many of these low energy cross \nsections are only known at the $10-40\\%$ level compared to deep inelastic \nscattering cross sections which have been measured to a few-percent at \nhigher energies.\n\nNeutrino scattering physics has ventured into a new era. Recent data \nsamples collected by both K2K and MiniBooNE have increased the \nstatistics of this low energy data collection by almost an order of magnitude.\nIn doing so, both experiments have added important new data on nuclear \ntargets ($^{16}O$, $^{12}C$) that are most relevant for use in neutrino\noscillation experiments. Higher energy data from the MINOS near \ndetector~\\cite{marchionni} that is just arriving will complement \nthis picture. In addition, much anticipated plans for future measurements \nat the MINERvA experiment as well as plans to use beams of antineutrinos, \nstopped pion sources, beta beams, and liquid argon detectors are also on \nthe horizon. These bring hope for rapid advancement in the field \nof neutrino scattering physics in the next decade or so. \\\\\n\nThe following sections briefly summarize the experimental results presented \nin this working group on these two fronts: recent measurements, and plans \nfor future experiments.\n\n\\section{Status of Current Measurements}\n\nSeveral exciting results made their debut at this workshop. \nThe first is a search for CC coherent $\\pi^+$ production using the fully \nactive scintillator detector in the K2K near detector \nensemble~\\cite{k2k-coherent}. In the case of coherent single pion production, \nthe neutrino scatters from the entire nucleus transferring negligble energy \nto the target. Hence, such interactions have the distinct signature of a\nnegligible amount of nuclear recoil and a very forward-emitted single pion.\nBoth CC $\\pi^+$ and NC $\\pi^0$ coherent production is possible. Many \nmeasurements of these cross sections exist, but mostly at higher \nenergies~\\cite{vilain}. Surprisingly, K2K observes no evidence \nfor CC coherent pion production and sets an upper limit on this cross section.\nThis intriguing result is the first experimental measurement of \nneutrino-induced coherent pion production at low energy. \\\\\n\nIn addition to their new coherent pion results, K2K also presented \nan update of their axial mass ($M_A$) fits from QE data in their near\ndetectors~\\cite{sakuda}. K2K's $M_A$ measurement is the first determination\nof this parameter on a water target and is part of an important effort\naimed at improving our current knowledge of QE cross sections and\nkinematics on nuclear targets. \\\\\n\nPreliminary CC scattering results from the MiniBooNE experiment were also \nshown for the first time. Using a simple requirement that \nevents contain two decay electrons in the final state, MiniBooNE isolates\na clean sample of CC $\\pi^+$ interactions in mineral oil. This sample \nconstitutes roughly five times more CC $\\pi^+$ data than all previous data \nsets combined. Combining this sample with comparably high statistics \nQE data taken from the same experiment, MiniBooNE reports the first \nmeasurement of the ratio of  CC $\\pi^+$\/QE cross sections at low energy \non a nuclear target~\\cite{jocelyn}. The measured ratio is roughly consistent \nwith current model predictions, given the large uncertainties in the\ntheory. In the future, MiniBooNE anticipates measurements of the NC $\\pi^0$ \nand coherent pion production cross sections, which will complement those \nfrom K2K. \\\\\n\nIn addition to neutrino measurements, electron scattering data is also\na necessary input towards improving low energy neutrino simulations.\nBecause neutrino oscillation experiments commonly employ heavy targets to \nensure high event yields, they rely on accurate nuclear model predictions. \nHigher statistics and more precise electron-nucleus scattering measurements \nare crucial for constraining and discriminating between such \nmodels (see for example~\\cite{sakuda-summary}). Dedicated low $Q^2$ electron\ndata, recently taken by the JUPITER experiment, comprise part of a \ncollaborative program linking the electron and neutrino physics communinities.\nA very preliminary look at these data was provided at this workshop with \nthe goal towards having $5\\%$ cross section measurements for use by the \nneutrino community in the near future~\\cite{bodek}.\n\n\\section{Future Initiatives}\n\nTalks on future neutrino cross section opportunities were featured in the\nsecond part of the session on experimental results. Antineutrino running at \nMiniBooNE was one such possibility~\\cite{morgan}. Given the scarcity of \nantineutrino cross section measurements at low energy, such data will be \nespecially important for future CP violation searches in the neutrino sector. \nIn addition, a proposal for bringing the fine-grained K2K SciBar detector to \nthe Fermilab Booster neutrino beamline was presented along with a discussion \nof its physics potential~\\cite{morgan}. If approved, this would allow \ncontinued measurement of $\\sim1$ GeV neutrino interactions making use of \na pre-existing fine-grained detector and intense low energy neutrino \nsource.  \\\\\n\nFurther, the recently approved MINERvA experiment will provide dedicated,\nhigh statistics neutrino scattering physics measurements starting in \n2008~\\cite{minerva}. The experiment features a fully active, \nfine-grained near detector placed in the newly commissioned NuMI beamline \nat Fermilab. This detector will uniquely allow measurements of neutrino \ninteractions across a very broad range of energies (the NuMI beam can run \nin low, medium, and high energy modes). MINERvA can perform an impressive \nlist of neutrino scattering studies with unprecedented detail, ranging from \nmeasurements of axial form factors, QE scattering, NC and CC resonance \nproduction, coherent pion production, strange particle production, deep \ninelastic scattering, and parton distributions~\\cite{minerva}. The \nincorporation of multiple nuclear targets will additionally allow a mapping \nof nuclear effects in neutrino interactions for the first time as well as \nprovide important information on the $A$ dependence of the coherent pion \ncross section. Given its broad potential for further improving our current \nknowledge, the construction and operation of this experiment is eagerly \nawaited. \\\\\n\nThe working group also expanded our discussion to include possible future\nneutrino scattering measurements at very low energies ($\\sim5-100$ MeV), the\nregion of interest for nuclear model builders and \nnuclear astrophysicists. Proposals for options to build a neutrino facility \nat the Spallation Neutron Source (SNS)~\\cite{stancu} and to make use of low \nenergy beta beams were discussed~\\cite{volpe}. Such measurements would help \nspan an important gap between between reactor and acclerator-based \nneutrino-nucleus measurements, where little data currently exists. \\\\\n\nFinally, the potential for neutrino scattering measurements using extremely \nhigh resolution liquid argon detectors was presented~\\cite{bonnie} along with \nsome new ideas on how to relate such measurements to those on more \nconventional targets~\\cite{lar}. The use of liquid argon could clearly \nallow more accurate determination of the various exclusive neutrino \nchannels and increase the precision of neutrino cross section measurements \nbeyond current capabilities.\n\n\\section{Questions for NuFact06}\n\nBoth the experimental and theoretical presentations in Working Group 2\nfocused primarily on surveying the current landscape, thereby providing\na sense of where we might be in the coming years. Future NuFact workshops \nmust now focus on where to go from here, specifically addressing which \nadditional neutrino scattering measurements are needed. Several questions\nalong these lines were presented at the workshop and are posed here:\n\n\\begin{itemize}\n  \\item After the complete set of anticipated results from the K2K, MiniBooNE,\n        and MINERvA experiments have arrived, what additional neutrino \n        scattering measurements can and should be made?\n  \\item Is there specific information needed by future neutrino oscillation\n        experiments that we will not already have in hand? Joint\n        Working Group 1 and 2 sessions should focus on this issue.\n  \\item What other neutrino beams should be investigated? Consider\n        the need for additional antineutrino measurements, \n        hadron production data, direct measurements of unoscillated $\\nue$ \n        cross sections, and narrow band beams for dedicated NC studies.\n  \\item What more can be gained from employing additional neutrino targets? \n        This includes new, high statistics, light target measurements \n        (off hydrogen and deuterium) and the use of polarized targets.\n  \\item Are there other detector technology improvements that should\n        be studied and pursued in addition to liquid argon detectors?\n\\end{itemize}\n\nNow that we have a better understanding of where we are headed in\nneutrino scattering physics in the short term, a concerted effort \nfocused on additional measurements, beams, and detectors we might \nneed beyond this is clearly necessary. \n\n\\section{Conclusions}\n\nNew data from K2K and MiniBooNE already help fill the gap in our\npresent knowledge of low energy neutrino cross sections. This workshop saw\nseveral new results from these experiments, including measurements \nin regions where there were previously no data. Future neutrino\noscillation experiments hoping for greater precision will require even\ntighter control of these cross sections. In this \nregard, we look forward to further advancement that can be achieved at \nthe MINERvA experiment and potentially from stopped pion sources, beta \nbeams, and liquid argon detectors. While past workshops (including this one)\nhave commonly concentrated on the current state of affairs in neutrino \nscattering physics, we encourage future workshops to project their focus \nfurther into the future.\n\n\\section{Acknowledgements}\n\nThe author thanks J. Morfin and fellow co-convenor, M. Sakuda, with whom \nit was a pleasure to organize this session. \nI would also like to thank all of the Working Group 2 speakers and \nparticipants, and extend sincere thanks to the NuFact05 organizers for \ntheir very generous travel support, which made it possible for many of \nour working group speakers to attend this workshop.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nDeep neural networks (DNNs) have shown excellent performance in challenging image segmentation tasks \\cite{long2015fully, DBLP:journals\/corr\/YuK15, jegou2017one, chen2017deeplab}. But their supervised training requires pixel labels for each training image. Manual annotation of pixels not only is laborious but also easily introduces label noise, especially in border regions. Compared to image classification, label noise is more common in image segmentation. So far, however, studies on the learning behavior of DNNs trained by noisy labels have focused primarily on classification. Is it necessary to accurately label each pixel for accurate segmentation? How is segmentation performance of DNNs influenced when trained by different types of noisy labels? Answering these questions not only will help us understand the role of labels in training of DNNs but also will provide insights into the learning behavior of DNNs.\n\n\n\n\\begin{table}[t]\n\\centering\n\\begin{tabular}{lll}\n\\toprule\n\tAbbreviation & Meaning & Description \\\\\n\\midrule\n    CL      &  Clean Label       & \\tabincell{l}{Ground truth labels \\\\ annotated by human \\\\ experts} \\\\ \n\\cline{1-3}\n    RCL    & \\tabincell{l}{Randomized \\\\ Clean Label}       & \\tabincell{l}{Randomly sampled \\\\ or flipped pixel labels \\\\ from ground truth}  \\\\\n\n\\cline{1-3}\n    PCL \t   & \\tabincell{l}{Perturbed \\\\ Clean Label} & \\tabincell{l}{Dilation\/Erosion\/\\\\ Skeleton of ground \\\\ truth} \\\\\n\\cline{1-3}\n    RL         & Random Label      & \\tabincell{l}{Randomly generated \\\\ pixel labels}    \\\\\n\\bottomrule\n\\end{tabular}\n\\caption{Different types of labels used in this study.}\n\\label{table1}\n\\end{table}\n\nDifferent types of noisy labels have been used to elucidate the learning behaviors of DNNs in image classification, including partially corrupted or randomly shuffled labels \\cite{DBLP:conf\/iclr\/ZhangBHRV17, DBLP:conf\/icml\/ArpitJBKBKMFCBL17}. In image segmentation, there are image-level and pixel-level label noises. Image-level label noise refer to erroneous semantic annotation of image objects. Pixel-level label noise refer to erroneous semantic annotation of image pixels. In this study, we focus on pixel-level label noise.  \n\nWe examine the performance of DNNs trained by four different types of labels, as summarized in Table 1 and shown in Figure 1 and Figure 4. To quantitatively analyze the segmentation performance of DNNs trained by these labels, we experiment on two representative segmentation models, U-Net \\cite{ronneberger2015u} and DeepLabv3+ \\cite{chen2018encoder}, with the same loss function (binary cross-entropy loss) and optimizer (stochastic gradient descent, SGD). Performance of DNNs trained by the labels ranks from the best to the worst as follows:\n\\begin{equation}\n\tCL \\approx RCL > PCL > RL\n\\end{equation}\nAs shown in Figure 1, when U-Net is trained with 45\\% of the labels randomly flipped (RCL) in binary segmentation, its performance remains largely the same as trained by the original ground truth (CL). Similar results on DeepLabv3+ are shown in Appendix A. These results indicate that DNNs learn structures hidden in the noisy labels rather than the pixel labels per se in their training for segmentation. We refer to these hidden structures as meta-structures. \n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.9\\columnwidth]{figure1}\n\\caption{Segmentation performance of DNNs trained by different types of labels. (A) Four types of training labels for an image of the endoplasmic reticulum from the ER dataset. CL: ground truth from manual annotation. RCL: each pixel label in CL is randomly flipped with a probability of 0.45. PCL: Skeleton of CL. RL: each pixel is randomly annotated as 1, i.e. foreground, with a probability of 0.5. (B) Testing dice scores during training. (C) Training loss of each batch optimization step.}\n\\label{fig1}\n\\end{figure}\n\nSimilar as observed in image classification in \\cite{DBLP:conf\/iclr\/ZhangBHRV17}, we also find that DNNs memorize random labels in segmentation since the training loss under RCL and RL quickly converges to a constant but not under CL and PCL (Figure 1C). Meanwhile, similar as observed in image classification in \\cite{DBLP:conf\/icml\/ArpitJBKBKMFCBL17}, we also find that before memorizing RL, DNNs prioritize learning real patterns first in segmentation because the dice score (Figure 1B, blue line) first fluctuates greatly then quickly drops to a low level. Since RL requires no annotation, this motivates us to develop an unsupervised segmentation model for binary segmentation. The model sets RL as the initial training label and iteratively updates the training label to RCL by incorporating meta-structure information.\n\n\\noindent \\textbf{Main Contributions}\n\n\\noindent The main research contributions of this study are as follows:\n\n\\begin{itemize}\n\\item We provide direct experimental evidence that DNNs learn implicit structures hidden in noisy labels in semantic segmentation. We name these implicit structures as meta-structures and model them mathematically as spatial density distributions. We show theoretically and experimentally how this model may quantify semantic meta-structure information in the noisy labels.\n\\item We have identified some basic properties of meta-structures. We find that DNNs trained with labels under different perturbations to the meta-structures exhibit consistently worse segmentation performance.\n\\item By incorporating meta-structure information, we have developed an unsupervised model for binary segmentation that outperforms the state-of-the-art unsupervised models and achieves remarkably competitive performance against supervised models.\n\\end{itemize}\n\n\\section{Methods}\n\n\n\\subsection{Generation of different Noisy Labels}\nWe synthesize different types of noisy labels as follows:\n\n\\textbf{(1) RCL:} We synthesize randomized clean label (RCL) by randomly sampling or flipping pixel labels in CL. For random sampling, we randomly select pixel labels of each class with a probability of $P_{sample}$ and exclude unsampled pixels from training. For random flipping, we randomly swapped a fraction of $P_{flip}$ of true labels with randomly selected labels from other classes. See Figure 2 and Figure 3 for examples of randomized binary labels and multi-class labels, respectively. \n\n\\textbf{(2) PCL:} We perform image dilation or erosion with a $3 \\times 3$ template or extract one-pixel-wide skeleton of CL. We refer to these types of noisy labels as perturbed clean labels (PCL). Examples of PCL are shown in Figure 4A.\n\n\\textbf{(3) RL:} We synthesize random label (RL) by randomly annotating pixel labels with a probability $P_{generate}$. RL can be considered as a strong perturbation of CL since it contains no information from CL. Examples of RL are shown in Figure 4B.\n\n\\subsection{Experiment Configuration}\nWe examine the learning behavior of two DNNs (U-Net and DeepLabv3+) in both binary-class and multi-class segmentation. For binary-class segmentation, we select fluorescence microscopy images of ER, MITO datasets \\cite{t2he-zn97-20} and the NUC dataset \\cite{caicedo2019nucleus}. For multi-class segmentation, we select natural images of Cityscapes dataset \\cite{cordts2016cityscapes}. Detailed information on the datasets and experimental configurations are provided in Appendix C.\n\n\n\n\\section{Experimental Evidence for Existence of Meta-Structures}\n\nWe examine the learning behavior of DNNs trained by RCL in both binary-class and multi-class segmentation.\n\n\\subsection{Existence of Meta-Structures}\n\nFigure 2 shows the testing dice scores of U-Net (Solid lines) and DeepLabv3+ (Dashed line) trained by binary RCL. Although the rate of convergence generally decreases under increase of label randomization, especially for DeepLabv3+, the final dice scores decrease only slightly and the maximum gap between final dice scores of CL and RCL is $7\\%$. Similar results are observed on other binary datasets. See Appendix C.3 for details.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.9\\columnwidth]{figure2}\n\\caption{Segmentation performance of DNNs trained by RCL generated by (A) random sampling and (B) random flipping on ER dataset. Solid lines: U-Net. Dashed lines: DeepLabv3+.}\n\\label{fig2}\n\\end{figure}\n\nFor multi-class labels, we train DeepLabv3 on the natural image dataset \\textit{Cityscapes} and compare with results of a previous study as our baseline \\cite{chen2017rethinking}. Due to a lack of previous results for comparison, we did not experiment with U-Net. Representative results are shown in Figure 3. Table 2 summarizes segmentation results of RCLs under different randomization probabilities. No degradation in segmentation performance measured in mean intersection over union (mIoU) is observed.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.9\\columnwidth]{figure3}\n\\caption{Segmentation performance of Deeplabv3 trained by noisy labels. Upper row: Examples of noisy-labels synthesized by random sampling($P_{sample}=0.1$) and random flipping($P_{flip}=0.9$). Lower row: Examples of segmented images in test set.}\n\\label{fig3}\n\\end{figure}\n\nTaking the results together, we reason that semantic information contained in CL is completely or largely preserved in RCL.\n\n\\begin{table}[t]\n\\centering\n\\begin{tabular}{lccc}\n\\toprule\nNoise Type & Noise Ratio & mIoU(\\%) \\\\\n\n\\midrule\n\n\\multirow{1}{*} {None}      & 0        & \\tabincell{c}{64.8 \\\\ (Chen, 2017)} \\\\ \n\\cline{1-3}\n   \\multirow{3}{*}{RS}    \t                  & 0.5\t\t  & 64.8   \\\\\n           \t\t\t\t\t\t\t\t~ \t  & 0.3\t\t  & 64.5   \\\\\n           \t\t\t\t\t\t\t\t~ \t  & 0.1 \t\t  & 64.6\t   \\\\ \t     \n           \t\t\t\t\t\t\t\t\n\\cline{1-3}\n   \\multirow{3}{*}{RF}                    \t  & 0.5\t\t  & 64.7   \\\\\n           \t\t\t\t\t\t\t\t~ \t  & 0.7\t\t  & 64.6   \\\\\n           \t\t\t\t\t\t\t\t~ \t  & 0.9 \t\t  & 64.7\t   \\\\                       \n           \t\t\t\t\t\t\t\t\n\n\\bottomrule\n\\end{tabular}\n\\caption{Segmentation performance of DeepLabv3 trained by randomly sampled (RS) and randomly flipped (RF) labels on Cityscapes.}\n\\label{table2}\n\\end{table}\n\n\\subsection{Meta-Structures vs. Pixel-level Labels}\n\nSo far, we have demonstrated that DNNs can learn segmentation from extremely noisy labels. However, it is unclear whether meta-structures or pixel-level labels contribute more to segmentation performance. In binary-class segmentation of fluorescence microscopy images, when $P_{flip}\\leq0.49$, the fraction of correctly annotated pixels ($\\geq$51\\%) still exceeds the fraction of incorrectly annotated pixels ($\\leq$49\\%). This raises the possibility that DNNs learn from major correct pixel-level labels rather than meta-structures. \n\nTo test this possibility, we generate entirely random labels, referred to as RL on ER dataset. Each pixel is randomly assigned to foreground with a certain generation probability. Sample image is shown in Figure 4B. Here, we set the generation probability as 0.1 and compare with the noisy label synthesized by random flipping with a probability $P_{flip} = 0.49$. While the randomly flipped labels still contain the meta-structures, the randomly generated labels do not. When we count the number of correctly annotated pixels using CL as the reference, we find that the pixel-level error rate of the randomly generated labels is around 31\\%, which is much lower than the error rate of randomly flipped labels (49\\%). If DNNs mainly learn from the pixel-level labels, the segmentation performance trained by RL would be better than the randomly flipped labels. \n\nHowever, segmentation performance of U-Net trained by RL is actually worse than randomly flipped labels (Figure 5A). Similar results using DeepLabv3+ are shown in Appendix C.4A. Together, these results further support that DNNs learn from meta-structures in labels rather than pixel-level labels per se in their supervised training for semantic segmentation.\n\n\\subsection{Summary}\n\nAlthough RCL is uncommon in real-world applications, it allowed us to discover this counterintuitive learning behavior that DNNs learn meta-structure rather than pixel-level labels per se in segmentation. Furthermore, we find that:\n\\begin{center}\n\t\\textit{DNNs trained by randomized labels that contain similar meta-structure information as the ground truth labels provide similar performance in semantic segmentation.}\n\\end{center}\n\nMathematical formulation and proof of this finding is presented later as Theorem 2 in the section \\textit{Theoretical Analysis of Meta-Structures}.\n\n\n\\section{Further Characterization of Meta-Structures}\n\nIn this section, we further characterize the properties of meta-structures by analyzing PCL and RL.\n\n\\subsection{Performance of DNNs Trained by PCL}\n Inaccurate boundaries are a common source of label noise in image segmentation. We simulate inaccurate boundaries using dilation, erosion and skeleton of CL, which we refer to as PCL (Table 1). We examine the influence of PCL on segmentation performance and the results for U-Net are shown in Figure 5B. Results for DeepLabv3+ and more smooth perturbation on boundaries using different combinations of dilation and erosion are shown in Appendix C.4B-C.\n \nDilation and erosion of CL lead to degradation in segmentation accuracy (Figure 5B) because of their perturbation of meta-structures. Further degradation in segmentation performance under training by one-pixel-wide skeleton is because of the loss of width information. We conclude that:\n \n\\begin{center}\n\t\\textit{Training of DNNs by labels with progressively stronger perturbation to the meta-structure exhibit progressively worse training performance}\n\\end{center}\n\nMathematical formulation and proof of this argument is presented later as Theorem 1 in the section \\textit{Theoretical Analysis of Meta-Structures}. \n\n\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[width=0.8\\textwidth]{figure4}\n\\caption{Different perturbations to ground truth (CL). (A): Examples of perturbed clean labels. (B): RL synthesized under different generation probabilities.}\n\\label{fig4}\n\\end{figure*}\n\n\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[width=0.8\\textwidth]{figure5}\n\\caption{Performance of DNNs trained under different perturbations to ground truth (CL). (A): Testing dice score trained by CL, RCL ($P_{flip}=0.49$) and RL ($P_{generate}=0.1$). (B): Testing dice score trained by CL and PCL. (C): Testing dice scores (left) and training loss (right) trained by RL.}\n\\label{fig5}\n\\end{figure*}\n\n\n\\subsection{Learning behavior of DNNs Trained by RL}\n\n \\citet{DBLP:conf\/iclr\/ZhangBHRV17} and \\citet{DBLP:conf\/icml\/ArpitJBKBKMFCBL17} have shown that DNNs learn simple patterns before fitting RL by memorization. This conclusion, however, is drawn in image classification. In our study, we investigate whether DNNs exhibit similar behavior in image segmentation. We generate RL under different generation probabilities (Figure 4B) from 0 to 0.5. We experiment on the ER dataset and show the results of U-Net in Figure 5C. \n\nFrom the testing results on ER dataset (Figure 5C, left panel), we find that the learning process consists of two stages. In the first stage, the dice scores fluctuate substantially and reach several high values, indicating that the U-Net keeps learning and is not yet strongly influenced by RL. In the second stage, the dice scores drop quickly then converge to a low value, indicating that the U-Net start memorizing as the generalization ability becomes worse. Meanwhile, we find that under a higher generation probability, the training loss converges more quickly (Figure 5C, right panel), indicating DNNs have a higher tendency to memorize. Similar results are observed on the MITO dataset. See Appendix C.4D for further details. Overall, the learning behavior of DNNs trained by RL in segmentation is consistent with the learning behavior of DNNs trained by RL in classification.\n\n\\subsection{Summary}\n\nCompared to CL and RCL, the PCL and RL contain different perturbations to meta-structures. We observe consistent degradation in segmentation performance.\n\n\n\n\n\\section{Unsupervised Binary-class Segmentation Based on Meta-Structures}\n\nHere we propose an unsupervised method for binary-class segmentation of biological microscopy images, which we name as iterative ground truth training (iGTT). Our goal is to provide an example of utilizing meta-structures in practice.\n\n\\subsection{Notation}\nGiven data pair $(X,Y) \\in \\mathbb{R}^{H \\times W}$, where $X$ denotes the input image, $Y$ denotes the binary-class labels with foreground and background labeled as 1 and 0, respectively. $H$ and $W$ denote image height and width, respectively. For our unsupervised method, we use the U-Net as the base model. The output of the final layer is denoted as $P$.\n\n\\subsection{Unsupervised Iteration Strategy}\nWe use fully black images to initialize the labels $Y^*$ then iteratively update it in following epochs. Specifically, for the $n_{th}$ epoch, the final layer of the image prediction score is $P^n=\\mathcal{F}(\\theta^{n}, X)$, where $\\mathcal{F}$ denotes the model and $\\theta$ are its parameters. We use sigmoid function in the final layer and only output the probability that belongs to the foreground for each pixel. We update the $\\theta$ based on current epoch labels $Y^*$. Then we refresh $Y^*$ according to the $P^{n}$ by generating a threshold set $T$ with $K$ thresholds as follows:\n\\begin{equation}\n\tT=\\{t\\mid t = p_{min} + k \\times \\Delta \\}\n\\end{equation}\n\\begin{equation}\n\t\\Delta =  \\frac{p_{max} - p_{min}}{K-1}\n\\end{equation}\nwhere $k=\\{0,1,\\dots,K-1\\}$. $p_{min}$ and $p_{max}$ denotes the minimum and maximum pixel value in $P^n$, respectively. $\\Delta$ denotes the interval between neighboring thresholds in threshold set $T$. \n\nBased on thresholds $t$ in $T$, we generate $K$ coarse segmentation images $S_k^n$ ($k=0,1,\\dots,K-1$) by thresholding $P^n$, where pixel $p$ is set to 1 if $p>t$. Next, we directly calculate the correlation between $P^n$ and $S_k^n$, then we find the most correlated segmentation image $\\widetilde{S}^n$ from $S_{k\\in\\{0,\\dots,K-1\\}}^n$ and consider $\\widetilde{S}^n$ as the optimal candidate labels for the next epoch of training.\n\nBecause we found most pixels $p_i^1$ of the output $P^1$ are close to 0 after first epoch training, if we used distance-based metrics for correlation calculation, the selected candidate labels $\\widetilde{S}^n$ will tend to be black images since the distance of pair ($p_i^1$ , 0) is smaller than ($p_i^1$ , 1). To address this issue, we follow \\cite{xu2019l_dmi} and select an information-theoretic noise-robust loss $\\mathcal{L}_{DMI}$ to measure the correlation between $P^n$ and $S^n_k$ labels as follows:\n\\begin{equation}\n\tCor(P^n,S_k^n) = \\mathcal{L}_{DMI}(P^n,S_k^n) = -\\log(\\left| det(Q_{(P^n \\parallel S_k^n)}) \\right|)\n\\end{equation}\nwhere $Q_{(P^n \\parallel S_k^n)}$ is the matrix form of the joint distribution over $P^n$ and $S_k^n$. To calculate the $Q_{(P^n \\parallel S_k^n)}$, we first resize the $P^{n} \\in \\mathbb{R}^{H \\times W}$ to $P_f \\in \\mathbb{R}^{1 \\times HW}$, then concatenate $P_f$ and $1-P_f$ to $\\mathcal{P} \\in \\mathbb{R}^{2 \\times HW}$. Meanwhile, we resize $S_k^n \\in \\mathbb{R}^{H \\times W}$ to $S_f \\in \\mathbb{R}^{1 \\times HW}$, then concatenate $S_f$ and $1-S_f$ to $\\mathcal{S} \\in \\mathbb{R}^{2 \\times HW}$. The $Q_{(P^n \\parallel S_k^n)}$ is defined by matrix multiplication as follows:\n\\begin{equation}\n\tQ_{(P^n \\parallel S_k^n)} = \\mathcal{P} \\mathcal{S}^{\\mathrm{T}}\n\\end{equation}\nBy calculating $Cor(P^n,S_k^n)$, we find the optimal candidate labels $\\widetilde{S}^n$ that has the highest mutual information with the prediction $P^n$. Then we send $\\widetilde{S}^n$ into an EMS module (i.e., extraction-of-meta-structure module, see next section) to extract the meta-structures $S^{meta}$. Finally, we update the label $Y^*$ by $S^{meta}$. The whole iteration strategy is summarized in Algorithm 1. The architecture of iGTT is shown in Appendix D.2.\n\n\\begin{algorithm}[tb]\n\\caption{Unsupervised Iteration Strategy}\n\\label{iteration strategy}\n\\textbf{Inputs}: $X_{train} \\in \\mathbb{R}$ \\\\\n\\textbf{Parameter}: threshold $t$, model $\\mathcal{F}$\n\\begin{algorithmic}[1]\n   \\STATE $n=1$\n   \\STATE $Y^*=0$\n   \\IF{$n<Maxiters$}\n   \\STATE $P^n = \\mathcal{F}(X_{train})$\n   \\STATE \\bfseries UPDATE $\\mathcal{F}$ with $Y^*$\n   \\FOR{$k \\in \\{0,1,...,K-1\\}$}\n   \\STATE $ S_k^n\\leftarrow\\left\\{\n             \\begin{array}{lr}\n             p=1 $ \\bfseries  if  $ p-t_k>0, p \\in P^n \\\\\n             p=0 $ \\bfseries  if  $ p-t_k<0, p \\in P^n\n             \\end{array} \\right.$\n   \\ENDFOR\n   \\STATE $\\widetilde{S}^n=argmin_{k \\in \\{0,...,K-1\\}}Cor(P^n, S_k^n)$\n   \\STATE $Y^*=S^{meta}=EMS(\\widetilde{S}^n)$\n   \\STATE $n=n+1$\n   \\ENDIF\n\\end{algorithmic}\n\\end{algorithm}\n\n\\subsection{Extraction-of-Meta-Structure Module}\n\nBecause of insufficient training, the segmented images $\\widetilde{S}^n$ by thresholding are coarse in early training steps. Thus we do not directly use $\\widetilde{S}^n$ as the next epoch training labels. However, the basic topology of objects are largely retained in $\\widetilde{S}^n$. We have demonstrated that DNNs trained by labels with correct topology structures (meta-structures) can achieve similar performance as ground-truth labels. Based on this, we design an extraction-of-meta-structure (EMS) module to further improve the quality of the pseudo labels.\n\nWe first extract the skeleton of the $\\widetilde{S}^n$, then we randomly shift every pixel in skeleton within a radius $r$ to disrupt the perturbation made by PCL. Since the randomly shift may move some pixel labels outside the target meta-structures, we follow a random sampling operation to filter out these pixel labels. The final pseudo label $S^{meta}$ generated by EMS module refines the meta-structures of the $\\widetilde{S}^n$. Then we directly update $Y^*$ as the $S^{meta}$ and use $Y^*$ for the next epoch training.\n\nTo optimize the model, we combine $\\mathcal{L}_{DMI}$ and $\\mathcal{L}_{IOU}$ \\cite{huang2019batching} to minimize the loss function $\\mathcal{L}$ for current epoch:\n\\begin{equation}\n\\begin{aligned}\n\t\\mathcal{L} = \\underbrace{-\\log (\\left|det(\\mathcal{P} \\mathcal{S}^{\\mathrm{T}})\\right|)}_{\\mathcal{L}_{DMI}} + \\\\ \\underbrace{(1 - \\frac{\\sum_{i=1}^{HW}p_iy_{i}^*}\n\t{\\sum_{i=1}^{HW}(p_i + y_{i}^* - p_iy_{i}^*) + \\epsilon})}_{\\mathcal{L}_{IOU}}\n\\end{aligned}\n\\end{equation}\nwhere $p_i$ and $y_i^*$ denote the $i_{th}$ pixel in image $P$ and $Y^*$, respectively. $\\epsilon$ is a smoothing coefficient to prevent the denominator from becoming zero.\n\n\\begin{table}[t]\n\\centering\n\\resizebox{.98\\columnwidth}{!}{\n\\begin{tabular}{cclll}\n\\toprule\nDataset & Model & DICE(\\%) & AUC(\\%) & ACC(\\%) \\\\\n\n\\midrule\n\\multirow{9}{*}{ER} & U-Net            & 85.99 & 97.09 & 91.09   \\\\\n & HRNet            & 86.07 & 97.17 & 91.18  \\\\\n & DeepLabv3+       & 81.66 & 94.80 & 87.67   \\\\\n \\cline{2-5}\n & AGT\t\t\t   & 76.23 & 82.63 & 85.19   \\\\\n & Otsu\t\t\t   & 69.47 & 76.76 & 84.76  \\\\\n & DFC\t\t\t   & 78.13 & 85.29 & 84.45  \\\\  \n & AC\t\t\t   & 73.11 & 87.86 & 81.41   \\\\\n & iGTT(w EMS)      & $78.84_{1.17}$ & $91.61_{1.04}$ & $85.41_{1.06} $ \\\\\n & iGTT(w\/o EMS) & $73.96_{0.97}$ & $84.53_{2.52}$ & $81.16_{1.03}$ \\\\\n\n\\bottomrule\n\\end{tabular}}\n\\caption{Segmentation performance of various models. Numbers in subscripts represent standard deviation.}\n\\label{table3}\n\\end{table}\n\n\\subsection{Segmentation Experiments}\n\nAs iGTT is customized for binary-class segmentation, we evaluate its performance on the ER, MITO \\cite{t2he-zn97-20} and NUC \\cite{caicedo2019nucleus} datasets. Refer to Appendix D.1 for further configuration details. \n\nTo compare with supervised methods, we select two commonly used DNNs (U-Net, DeepLabv3+) and a state-of-the-art model HRNet \\cite{wang2020deep}. To compare with unsupervised methods, we select adaptive gaussian thresholding (AGT), Otsu, and two state-of-the-arts methods which including Autoregressive Clustering (AC) \\cite{ouali2020autoregressive} and Differentiable Feature Clustering (DFC) \\cite{kim2020unsupervised}. We use DICE(dice scores), AUC(area under curve) and ACC(accuracy) as the performance metrics. To reduce the effects brought by randomization, we trained our model 10 times and calculate the mean and standard deviation. \n\nSegmentation results on ER are summarized in Table 3. See Appendix D.3 for results on MITO and NUC datasets. For supervised models, HRNet achieves the best performance. The DeepLabv3+ performs worse than U-Net and HRNet. For unsupervised methods, iGTT achieves the best performance. Meanwhile, we find that iGTT achieves competitive performance when comparing with the other three supervise models. Moreover, we find that using EMS module improves the final segmentation performance, indicating that EMS indeed refines the candidate labels. Overall, our model effectively narrows the gap between supervised learning and unsupervised learning by effectively utilizing the implicit meta-structures in noisy-labels. Examples of testing segmented images on all datasets are shown in Appendix D.4 and D.5.\n\n\n\\section{Theoretical Analysis of Meta-Structures}\nWe model meta-structures of labels based on the theorem of spatial point analysis \\cite{baddeley2015spatial,diggle2013statistical,illian2008statistical}. Specifically, we use $P(y^*=j|y=i)$ to denote the probability of flipping ground-truth pixel label $y$ in class $i \\in \\{1,\\dots,M \\}$ to the noisy pixel label $y^*$ in class $j \\in \\{1,\\dots,M \\}$, where $M$ denotes the number of semantic classes. For RCL and RL, we can build noise transition matrix $Q_{y^*|y}$ based on $P(y^*=j|y=i)$. Examples of $Q_{y^*|y}$ are shown Figure 6. \n\n\nFor binary-class semantic labels, we treat the foreground pixels ($y^*=1$) as the spatial data point. For multiple-class semantic labels, we disassemble multi-class labels as multiple binary-class labels and separately analysis the spatial density distributions of each class. Specifically, We treat the pixel $x^m$ whose label $y^*=m \\in \\{1,\\dots,M \\}$ as the spatial data point and view other pixels $(y^* \\neq m)$ as background by annotating as 0. We use $N$ to denote the number of spatial data points, $h$ to denote the bandwith (radius) of the search area $S$, and $K$ to denote the kernel function.\n\n\\textbf{Definition.} We define the Meta-structures of a label (MS) as a set of sematic classes $MS=\\{O_1,...,O_m\\}$, where each class $O_i=\\{x^m|x^m \\sim f_i (x^m) \\}$ is composed of pixels $x^m$ that are drawn from the same underlying spatial density distribution $f_i (x^m)$.\n\n\\textbf{Lemma 1.} When pixels $x^m$ are treated as data points, the density distribution $f_i(x^m)$ in random noisy labels can be calculated as follows: \n\\begin{equation}\n\tf_i(x^m)=\\{ \\frac{1}{2Nh} * \\sum_{j=1}^M P(y^*=m|y=j)*S_j \\} \\pm \\delta\n\\end{equation}\nwhere $\\delta$ denotes sampling errors, which are a constant, $S_j=S \\cap O_j$ is the area that corresponds to $j_{th}$ semantic class $O_j$ within the search area $S$.\n\n\\textbf{Proof:} Based on the theorem of spatial point analysis, we use kernel function $K$ to estimate the density distribution $f_i(x^m)$ by counting $x^m$ within a search area as follows:\n\\begin{equation}\n\tf_i(x^m)= \\frac{1}{2Nh} \\sum_{k=1}^N K(x-h \\le x_k^m \\le x+h)\n\\end{equation}\nNote that for RCL, the number of counted data points within the search area can be formulated as follows: \n\\begin{equation}\n\t\\sum_{k=1}^N K(x-h \\le x_k^m \\le x+h) = \\sum_{j=1}^M P(y^*=m|y=j)*S_j \\pm \\delta\n\\end{equation} \nBecause $N$, $h$, $\\delta$ and $S_j$ are all constant, $f_i(x^m)$ is only dependent on flipping probability $P(y^*=m|y=j)$. \\textit{QED} (See further details in Appendix B.1)\n\n\\textbf{Lemma 2.} The number of semantic classes $D$ in random labels equals the rank $R$ of noise transition matrix $Q_{y^*|y}$: $D=R$.\n\n\\textbf{Proof:}  If $R<M$, there exits $(M-R+1)$ linearly correlative columns and $P(y^*=j|y=1)=...=P(y^*=j|y=M-R)$. Based on Lemma 1, $f_i(x^m)$ is almost the same constant $C$ within the area $A=\\{y=1\\} \\cup ... \\cup \\{y=M-R\\}$, indicating that $(M-R+1)$ density distributions of the area $A$ are fused into one. Based on the definition that semantic class is composed of pixels that have similar density distributions, the number of semantic classes $D=M-(M-R)=R$.\n\nIf $R=M$, for each column of $Q_{y*|y}$ there at least exists $P(y^*=m|y=p) \\neq P(y^*=m|y=q)$ and the $f_i(x^m)$ are different within the area $A=\\{y=p\\}\\cup\\{y=q\\}$. Thus distribution differences on all classes exist and the number of semantic classes $D=M=R$. In summary, $D=R$. \\textit{QED}\n\n\\textbf{Theorem 1.} For PCL, more bias on boundary pixels of ground-truth more perturbation on meta-structures.\n\n\\textbf{Proof:} According to the definition of meta-structure, the semantic boundaries between $Y^*$ and $Y$ are different since more bias on boundary pixels in PCL will lead to less interaction of semantic class $O_i$ between PCL and CL, and therefore more perturbation on meta-structures. \\textit{QED} \n\n\\textbf{Theorem 2.} If RCL $Y^*$ is synthesized by $Q_{y^*|y}$ and the rank of $Q_{y^*|y}$ is full, the noisy labels $Y^*$ have the similar semantic information as ground-truth $Y$.\n\n\\textbf{Proof:} Based on Lemma 2, if $R(Q_{y^*|y})=M$, the meta-structure of $Y^*$ contains the complete semantic classes as $Y$: $MS(Y^*) = MS(Y)= \\{ O_1,\\dots,O_m \\}$. Furthermore, the location of difference between density distributions is unchanged, indicating semantic areas of different classes are also unchanged. Thus, all classes in $Y^*$ keep distinguishability like $Y$ and both of them contain similar semantic information. \\textit{QED}\n\n We also demonstrate Theorem 2 by experiments. We generate a simplified binary image of (256\u00d7256) that has a circle within a rectangle as shown in Figure 6 (first column). For the noisy labels (RCL) whose rank equals 2 (second column), its density distribution estimated by kernel function exhibits similar patterns as the ground-truth, indicating they have similar meta-structures. However, for the noisy labels (RL) whose rank equals 1 (third column), its density distribution is randomized, indicating that two semantic classes are fused together. More experiment results are shown in Appendix B.2.\n \n\n\\textbf{Summary.} Here we provide a direct mathematical definition of meta-structures. This definition theoretically and experimentally explains the key observed results in our segmentation experiments.  \n\n \\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.9\\columnwidth]{figure6}\n\\caption{Estimation of density distribution of noisy labels with different ranks (from left to right: ground truth, RCL, RL). NTM: noise transition matrix. KDE: kernel density estimation.}\n\\label{fig6}\n\\end{figure}\n\n\\section{Related Work}\n\n\\textbf{Training with noisy labels.} To explore the generalization properties of DNNs trained with noisy labels, \\citet{DBLP:conf\/iclr\/ZhangBHRV17} and \\citet{DBLP:conf\/icml\/ArpitJBKBKMFCBL17} performed a series of experiments and demonstrated that DNNs can easily memorize random labels but with poor generalization. This phenomenon contradicts traditional statistical learning theory \\cite{vapnik1999overview, bartlett2005local} and has attracted a large number of studies on how to mitigate the negative influence of noisy labels in deep learning.\n\nIn image classification, many studies have tried to propose noise-robust loss functions \\cite{manwani2013noise, masnadi2008design, brooks2011support, van2015learning, ghosh2017robust, zhang2018generalized, xu2019l_dmi}. Several works focus on designing custom architectures for deep neural networks \\cite{jiang2018mentornet, han2018co}. Transfer learning has also been applied \\cite{lee2018cleannet}. Some other studies proposed multi-tasks frameworks to estimates true labels \\cite{veit2017learning, tanaka2018joint, yi2019probabilistic, li2020dividemix}. Although most of these methods performed well in image classification tasks, their performance in image segmentation remains unknown.\n\nIn semantic segmentation, \\citet{min2019two} proposed to weaken the influence of back-propagated gradients caused by incorrect labels based on mutual attention. \\citet{shu2019lvc} proposed to leverage local visual cues to automatically correct label errors. Several studies also proposed semi-supervised or unsupervised methods \\cite{lu2016learning, li2019supervised, navlakha2013unsupervised, zheng2020rectifying}.\n\nAll these studies focus on improving the generalization ability of DNNs trained with noisy labels. However, an in-depth understanding of the fundamental roles of labels in training remains lacking. We address this deficiency by focusing on the noisy labels and examine their influence on image segmentation directly and systemically.  \n\n\\textbf{Unsupervised segmentation.} Some works try to segment images by learning pixel representation in a self-supervised setting \\cite{hwang2019segsort, zhang2020self}. However, they still relied on initialization from other annotated datasets. A limited number of studies address image segmentation in a fully unsupervised way. Methods have been proposed to maximize the mutual information between augmented views \\cite{ouali2020autoregressive, ji2019invariant}. A DNN architecture consisting of convolutional filters for feature extraction and differentiable processes for feature clustering has also been proposed \\cite{kim2020unsupervised}. Overall, performance of unsupervised segmentation methods remains to be further improved.\n\n\n\\section{Conclusion}\n\nIn this study, we examine the learning behavior of DNNs trained by different types of pixel-level noisy labels in semantic segmentation and provide direct experimental evidence and theoretical proof that DNNs learn meta-structures from noisy labels. The unsupervised segmentation model we have developed provides an example on how to utilize the meta-structures in practice. However, our study also has its limitations. In particular, our model of meta-structures remains to be further developed, and utilization of meta-structures remains to be expanded to other applications such as multi-class segmentation. Despite these limitations, the learning behavior of DNNs revealed in this study provides new insight into what and how DNNs learn from noisy labels to segment images.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}