We introduce a notion of usable information contained in the representation learned by a deep network, and use it to study how optimal representations for the task emerge during training, and how they adapt to different tasks. We use this to characterize the transient dynamics of deep neural networks on perceptual decision-making tasks inspired by neuroscience literature. In particular, we show that both the random initialization and the implicit regularization from Stochastic Gradient Descent play an important role in learning minimal sufficient representations for the task. If the network is not randomly initialized, we show that the training may not recover an optimal representation, increasing the chance of overfitting. An important open question for the theory of deep learning is why highly overparametrized neural networks learn solutions that generalize well even though models can in principle memorize the entire training set. Some have speculated that neural networks learn minimal but sufficient representations of the input through implicit regularization of Stochastic Gradient Descent (SGD) (Shwartz-Ziv & Tishby, 2017; Achille & Soatto, 2018), and that the minimality of the representations relates to generalizability. Followup work has disputed some of these claims (Saxe et al., 2018), leading to an ongoing debate on the optimality of representations and how they are learned during training. Here, we design a simple task to empirically study how representations are formed during training, and how implicit regularization from SGD and initializations affect the resulting representations in deep networks. We then validate these results on a variant of an MNIST classification task to assess how SGD affects the minimality of representations. Investigations into the optimality of representations have typically used information-theoretic reasoning.

While nonparametric density estimators often perform well on low dimensional data, their performance can suffer when applied to higher dimensional data, owing presumably to the curse of dimensionality. One technique for avoiding this is to assume no dependence between features and that the data are sampled from a separable density. This allows one to estimate each marginal distribution independently thereby avoiding the slow rates associated with estimating the full joint density. This is a strategy employed in naive Bayes models and is analogous to estimating a rank-one tensor. In this paper we investigate whether these improvements can be extended to other simplified dependence assumptions which we model via nonnegative tensor decompositions. In our central theoretical results we prove that restricting estimation to low-rank nonnegative PARAFAC or Tucker decompositions removes the dimensionality exponent on bin width rates for multidimensional histograms. These results are validated experimentally with high statistical significance via direct application of an existing nonnegative tensor factorization to histogram estimators.

Graph neural networks (GNNs) in general, and graph convolutional networks (GCN) in particular, often rely on low-pass graph filters to incorporate geometric information in the form of local smoothness over neighboring nodes. While this approach performs well on a surprising number of standard benchmarks, the efficacy of such models does not translate consistently to more complex domains, such as graph data in the biochemistry domain. We argue that these more complex domains require priors that encourage learning of band-pass and high-pass features rather than oversmoothed signals of standard GCN architectures. Here, we propose an alternative GNN architecture, based on a relaxation of recently proposed geometric scattering transforms, which consists of a cascade of graph wavelet filters. Our learned geometric scattering (LEGS) architecture adaptively tunes these wavelets and their scales to encourage band-pass features to emerge in learned representations. This results in a simplified GNN with significantly fewer learned parameters compared to competing methods. We demonstrate the predictive performance of our method on several biochemistry graph classification benchmarks, as well as the descriptive quality of its learned features in biochemical graph data exploration tasks. Our results show that the proposed LEGS network matches or outperforms popular GNNs, as well as the original geometric scattering construction, while also retaining certain mathematical properties of its handcrafted (nonlearned) design.

Data-driven sparse system identification becomes the general framework for a wide range of problems in science and engineering. It is a problem of growing importance in applied machine learning and artificial intelligence algorithms. In this work, we developed the Entropic Regression Software Package (ERFit), a MATLAB package for sparse system identification using the entropic regression method. The code requires minimal supervision, with a wide range of options that make it adapt easily to different problems in science and engineering. The ERFit is available at https://github.com/almomaa/ERFit-Package

We consider low-distortion embeddings for subspaces under \emph{entrywise nonlinear transformations}. In particular we seek embeddings that preserve the norm of all vectors in a space $S = \{y: y = f(x)\text{ for }x \in Z\}$, where $Z$ is a $k$-dimensional subspace of $\mathbb{R}^n$ and $f(x)$ is a nonlinear activation function applied entrywise to $x$. When $f$ is the identity, and so $S$ is just a $k$-dimensional subspace, it is known that, with high probability, a random embedding into $O(k/\epsilon^2)$ dimensions preserves the norm of all $y \in S$ up to $(1\pm \epsilon)$ relative error. Such embeddings are known as \emph{subspace embeddings}, and have found widespread use in compressed sensing and approximation algorithms. We give the first low-distortion embeddings for a wide class of nonlinear functions $f$. In particular, we give additive $\epsilon$ error embeddings into $O(\frac{k\log (n/\epsilon)}{\epsilon^2})$ dimensions for a class of nonlinearities that includes the popular Sigmoid SoftPlus, and Gaussian functions. We strengthen this result to give relative error embeddings under some further restrictions, which are satisfied e.g., by the Tanh, SoftSign, Exponential Linear Unit, and many other `soft' step functions and rectifying units. Understanding embeddings for subspaces under nonlinear transformations is a key step towards extending random sketching and compressing sensing techniques for linear problems to nonlinear ones. We discuss example applications of our results to improved bounds for compressed sensing via generative neural networks.

Contrastive approaches to representation learning have recently shown great promise. In contrast to generative approaches, these contrastive models learn a deterministic encoder with no notion of uncertainty or confidence. In this paper, we introduce a simple approach based on "contrasting distributions" that learns to assign uncertainty for pretrained contrastive representations. In particular, we train a deep network from a representation to a distribution in representation space, whose variance can be used as a measure of confidence. In our experiments, we show that this deep uncertainty model can be used (1) to visually interpret model behavior, (2) to detect new noise in the input to deployed models, (3) to detect anomalies, where we outperform 10 baseline methods across 11 tasks with improvements of up to 14% absolute, and (4) to classify out-of-distribution examples where our fully unsupervised model is competitive with supervised methods. The success of supervised learning relies heavily on large datasets with semantic annotations. But as the prediction tasks we are interested in become increasingly complex -- such as applications in radiology (Irvin et al., 2019), law (Wang et al., 2013), and autonomous driving (Maurer et al., 2016) -- the expense and difficulty of annotation quickly grows to be unmanageable. As such, learning useful representations without human annotation is an important, longstanding problem. These "unsupervised" approaches largely span two categories: generative and discriminative. Generative models seek to capture the data density using ideas from approximate Bayesian inference (Hinton et al., 2006; Kingma & Welling, 2013; Rezende et al., 2014) and game theory (Goodfellow et al., 2014; Dumoulin et al., 2016).

Nonnegative matrix factorization (NMF) is a standard linear dimensionality reduction technique for nonnegative data sets. In order to measure the discrepancy between the input data and the low-rank approximation, the Kullback-Leibler (KL) divergence is one of the most widely used objective function for NMF. It corresponds to the maximum likehood estimator when the underlying statistics of the observed data sample follows a Poisson distribution, and KL NMF is particularly meaningful for count data sets, such as documents or images. In this paper, we first collect important properties of the KL objective function that are essential to study the convergence of KL NMF algorithms. Second, together with reviewing existing algorithms for solving KL NMF, we propose three new algorithms that guarantee the non-increasingness of the objective function. We also provide a global convergence guarantee for one of our proposed algorithms. Finally, we conduct extensive numerical experiments to provide a comprehensive picture of the performances of the KL NMF algorithms.

Understanding the relationships between biomedical terms like viruses, drugs, and symptoms is essential in the fight against diseases. Many attempts have been made to introduce the use of machine learning to the scientific process of hypothesis generation (HG), which refers to the discovery of meaningful implicit connections between biomedical terms. However, most existing methods fail to truly capture the temporal dynamics of scientific term relations and also assume unobserved connections to be irrelevant (i.e., in a positive-negative (PN) learning setting). To break these limits, we formulate this HG problem as future connectivity prediction task on a dynamic attributed graph via positive-unlabeled (PU) learning. Then, the key is to capture the temporal evolution of node pair (term pair) relations from just the positive and unlabeled data. We propose a variational inference model to estimate the positive prior, and incorporate it in the learning of node pair embeddings, which are then used for link prediction. Experiment results on real-world biomedical term relationship datasets and case study analyses on a COVID-19 dataset validate the effectiveness of the proposed model.

We consider the classical setting of optimizing a nonsmooth Lipschitz continuous convex function over a convex constraint set, when having access to a (stochastic) first-order oracle (FO) for the function and a projection oracle (PO) for the constraint set. It is well known that to achieve $\epsilon$-suboptimality in high-dimensions, $\Theta(\epsilon^{-2})$ FO calls are necessary. This is achieved by the projected subgradient method (PGD). However, PGD also entails $O(\epsilon^{-2})$ PO calls, which may be computationally costlier than FO calls (e.g. nuclear norm constraints). Improving this PO calls complexity of PGD is largely unexplored, despite the fundamental nature of this problem and extensive literature. We present first such improvement. This only requires a mild assumption that the objective function, when extended to a slightly larger neighborhood of the constraint set, still remains Lipschitz and accessible via FO. In particular, we introduce MOPES method, which carefully combines Moreau-Yosida smoothing and accelerated first-order schemes. This is guaranteed to find a feasible $\epsilon$-suboptimal solution using only $O(\epsilon^{-1})$ PO calls and optimal $O(\epsilon^{-2})$ FO calls. Further, instead of a PO if we only have a linear minimization oracle (LMO, a la Frank-Wolfe) to access the constraint set, an extension of our method, MOLES, finds a feasible $\epsilon$-suboptimal solution using $O(\epsilon^{-2})$ LMO calls and FO calls---both match known lower bounds, resolving a question left open since White (1993). Our experiments confirm that these methods achieve significant speedups over the state-of-the-art, for a problem with costly PO and LMO calls.

The variational auto-encoder (VAE) is a deep latent variable model that has two neural networks in an autoencoder-like architecture; one of them parameterizes the model's likelihood. Fitting its parameters via maximum likelihood is challenging since the computation of the likelihood involves an intractable integral over the latent space; thus the VAE is trained instead by maximizing a variational lower bound. Here, we develop a maximum likelihood training scheme for VAEs by introducing unbiased gradient estimators of the log-likelihood. We obtain the unbiased estimators by augmenting the latent space with a set of importance samples, similarly to the importance weighted auto-encoder (IWAE), and then constructing a Markov chain Monte Carlo (MCMC) coupling procedure on this augmented space. We provide the conditions under which the estimators can be computed in finite time and have finite variance. We demonstrate experimentally that VAEs fitted with unbiased estimators exhibit better predictive performance on three image datasets.