Characterizing dynamical systems given limited measurements is a common challenge throughout the physical and biological sciences. However, this task is challenging, especially due to transient variability in systems with equivalent long-term dynamics. We address this by introducing smooth prototype equivalences (SPE), a framework that fits a diffeomorphism using normalizing flows to distinct prototypes - simplified dynamical systems that define equivalence classes of behavior. SPE enables classification by comparing the deformation loss of the observed sparse, high-dimensional measurements to the prototype dynamics. Furthermore, our approach enables estimation of the invariant sets of the observed dynamics through the learned mapping from prototype space to data space. Our method outperforms existing techniques in the classification of oscillatory systems and can efficiently identify invariant structures like limit cycles and fixed points in an equation-free manner, even when only a small, noisy subset of the phase space is observed. Finally, we show how our method can be used for the detection of biological processes like the cell cycle trajectory from high-dimensional single-cell gene expression data.

The ability to fool deep learning classifiers with tiny perturbations of the input has lead to the development of adversarial training in which the loss with respect to adversarial examples is minimized in addition to the training examples. While adversarial training improves the robustness of the learned classifiers, the procedure is computationally expensive, sensitive to hyperparameters and may still leave the classifier vulnerable to other types of small perturbations. In this paper we analyze the adversarial robustness of the 1 Nearest Neighbor (1NN) classifier and compare its performance to adversarial training. We prove that under reasonable assumptions, the 1 NN classifier will be robust to {\em any} small image perturbation of the training images and will give high adversarial accuracy on test images as the number of training examples goes to infinity. In experiments with 45 different binary image classification problems taken from CIFAR10, we find that 1NN outperform TRADES (a powerful adversarial training algorithm) in terms of average adversarial accuracy. In additional experiments with 69 pretrained robust models for CIFAR10, we find that 1NN outperforms almost all of them in terms of robustness to perturbations that are only slightly different from those seen during training. Taken together, our results suggest that modern adversarial training methods still fall short of the robustness of the simple 1NN classifier. our code can be found at https://github.com/amirhagai/On-Adversarial-Training-And-The-1-Nearest-Neighbor-Classifier

Modern GANs achieve remarkable performance in terms of generating realistic and diverse samples. This has led many to believe that ``GANs capture the training data manifold''. In this work we show that this interpretation is wrong. We empirically show that the manifold learned by modern GANs does not fit the training distribution: specifically the manifold does not pass through the training examples and passes closer to out-of-distribution images than to in-distribution images. We also investigate the distribution over images implied by the prior over the latent codes and study whether modern GANs learn a density that approximates the training distribution. Surprisingly, we find that the learned density is very far from the data distribution and that GANs tend to assign higher density to out-of-distribution images. Finally, we demonstrate that the set of images used to train modern GANs are often not part of the typical set described by the GANs' distribution.

Papers published at the Neural Information Processing Systems Conference.

A longstanding problem in machine learning is to find unsupervised methods that can learn the statistical structure of high dimensional signals. In recent years, GANs have gained much attention as a possible solution to the problem, and in particular have shown the ability to generate remarkably realistic high resolution sampled images. At the same time, many authors have pointed out that GANs may fail to model the full distribution ("mode collapse") and that using the learned models for anything other than generating samples may be very difficult. In this paper, we examine the utility of GANs in learning statistical models of images by comparing them to perhaps the simplest statistical model, the Gaussian Mixture Model. First, we present a simple method to evaluate generative models based on relative proportions of samples that fall into predetermined bins. Unlike previous automatic methods for evaluating models, our method does not rely on an additional neural network nor does it require approximating intractable computations. Second, we compare the performance of GANs to GMMs trained on the same datasets. While GMMs have previously been shown to be successful in modeling small patches of images, we show how to train them on full sized images despite the high dimensionality. Our results show that GMMs can generate realistic samples (although less sharp than those of GANs) but also capture the full distribution, which GANs fail to do. Furthermore, GMMs allow efficient inference and explicit representation of the underlying statistical structure. Finally, we discuss how GMMs can be used to generate sharp images.

A longstanding problem in machine learning is to find unsupervised methods that can learn the statistical structure of high dimensional signals. In recent years, GANs have gained much attention as a possible solution to the problem, and in particular have shown the ability to generate remarkably realistic high resolution sampled images. At the same time, many authors have pointed out that GANs may fail to model the full distribution ("mode collapse") and that using the learned models for anything other than generating samples may be very difficult. In this paper, we examine the utility of GANs in learning statistical models of images by comparing them to perhaps the simplest statistical model, the Gaussian Mixture Model. First, we present a simple method to evaluate generative models based on relative proportions of samples that fall into predetermined bins. Unlike previous automatic methods for evaluating models, our method does not rely on an additional neural network nor does it require approximating intractable computations. Second, we compare the performance of GANs to GMMs trained on the same datasets. While GMMs have previously been shown to be successful in modeling small patches of images, we show how to train them on full sized images despite the high dimensionality. Our results show that GMMs can generate realistic samples (although less sharp than those of GANs) but also capture the full distribution, which GANs fail to do. Furthermore, GMMs allow efficient inference and explicit representation of the underlying statistical structure. Finally, we discuss how GMMs can be used to generate sharp images.

The likelihood function of a finite mixture model is a non-convex function with multiple local maxima and commonly used iterative algorithms such as EM will converge to different solutions depending on initial conditions. In this paper we ask: is it possible to assess how far we are from the global maximum of the likelihood? Since the likelihood of a finite mixture model can grow unboundedly by centering a Gaussian on a single datapoint and shrinking the covariance, we constrain the problem by assuming that the parameters of the individual models are members of a large discrete set (e.g. estimating a mixture of two Gaussians where the means and variances of both Gaussians are members of a set of a million possible means and variances). For this setting we show that a simple upper bound on the likelihood can be computed using convex optimization and we analyze conditions under which the bound is guaranteed to be tight. This bound can then be used to assess the quality of solutions found by EM (where the final result is projected on the discrete set) or any other mixture estimation algorithm. For any dataset our method allows us to find a finite mixture model together with a dataset-specific bound on how far the likelihood of this mixture is from the global optimum of the likelihood

In recent years, approaches based on machine learning have achieved state-of-the-art performance on image restoration problems. Successful approaches include both generative models of natural images as well as discriminative training of deep neural networks. Discriminative training of feed forward architectures allows explicit control over the computational cost of performing restoration and therefore often leads to better performance at the same cost at run time. In contrast, generative models have the advantage that they can be trained once and then adapted to any image restoration task by a simple use of Bayes' rule. In this paper we show how to combine the strengths of both approaches by training a discriminative, feed-forward architecture to predict the state of latent variables in a generative model of natural images. We apply this idea to the very successful Gaussian Mixture Model (GMM) of natural images. We show that it is possible to achieve comparable performance as the original GMM but with two orders of magnitude improvement in run time while maintaining the advantage of generative models.

Motivated by recent progress in natural image statistics, we use newly available datasets with ground truth optical flow to learn the local statistics of optical flow and rigorously compare the learned model to prior models assumed by computer vision optical flow algorithms. We find that a Gaussian mixture model with 64 components provides a significantly better model for local flow statistics when compared to commonly used models. We investigate the source of the GMMs success and show it is related to an explicit representation of flow boundaries. We also learn a model that jointly models the local intensity pattern and the local optical flow. In accordance with the assumptions often made in computer vision, the model learns that flow boundaries are more likely at intensity boundaries. However, when evaluated on a large dataset, this dependency is very weak and the benefit of conditioning flow estimation on the local intensity pattern is marginal.

Recently, researchers have demonstrated that loopy belief propagation - the use of Pearls polytree algorithm IN a Bayesian network WITH loops OF error- correcting codes.The most dramatic instance OF this IS the near Shannon - limit performance OF Turbo Codes codes whose decoding algorithm IS equivalent TO loopy belief propagation IN a chain - structured Bayesian network. IN this paper we ask : IS there something special about the error - correcting code context, OR does loopy propagation WORK AS an approximate inference schemeIN a more general setting? We compare the marginals computed using loopy propagation TO the exact ones IN four Bayesian network architectures, including two real - world networks : ALARM AND QMR.We find that the loopy beliefs often converge AND WHEN they do, they give a good approximation TO the correct marginals.However,ON the QMR network, the loopy beliefs oscillated AND had no obvious relationship TO the correct posteriors. We present SOME initial investigations INTO the cause OF these oscillations, AND show that SOME simple methods OF preventing them lead TO the wrong results.