RACE: Sub-Linear Memory Sketches for Approximate Near-Neighbor Search on Streaming Data

We present the first sublinear memory sketch which can be queried to find the $v$ nearest neighbors in a dataset. Our online sketching algorithm can compress an $N$-element dataset to a sketch of size $O(N^b \log^3{N})$ in $O(N^{b+1} \log^3{N})$ time, where $b < 1$ when the query satisfies a data-dependent near-neighbor stability condition. We achieve data-dependent sublinear space by combining recent advances in locality sensitive hashing (LSH)-based estimators with compressed sensing. Our results shed new light on the memory-accuracy tradeoff for near-neighbor search. The techniques presented reveal a deep connection between the fundamental compressed sensing (or heavy hitters) recovery problem and near-neighbor search, leading to new insight for geometric search problems and implications for sketching algorithms.

Representing Formal Languages: A Comparison Between Finite Automata and Recurrent Neural Networks

We investigate the internal representations that a recurrent neural network (RNN) uses while learning to recognize a regular formal language. Specifically, we train a RNN on positive and negative examples from a regular language, and ask if there is a simple decoding function that maps states of this RNN to states of the minimal deterministic finite automaton (MDFA) for the language. Our experiments show that such a decoding function indeed exists, and that it maps states of the RNN not to MDFA states, but to states of an {\em abstraction} obtained by clustering small sets of MDFA states into "superstates". A qualitative analysis reveals that the abstraction often has a simple interpretation. Overall, the results suggest a strong structural relationship between internal representations used by RNNs and finite automata, and explain the well-known ability of RNNs to recognize formal grammatical structure.

Adaptive Estimation for Approximate k-Nearest-Neighbor Computations

Algorithms often carry out equally many computations for "easy" and "hard" problem instances. In particular, algorithms for finding nearest neighbors typically have the same running time regardless of the particular problem instance. In this paper, we consider the approximate k-nearest-neighbor problem, which is the problem of finding a subset of O(k) points in a given set of points that contains the set of k nearest neighbors of a given query point. We propose an algorithm based on adaptively estimating the distances, and show that it is essentially optimal out of algorithms that are only allowed to adaptively estimate distances. We then demonstrate both theoretically and experimentally that the algorithm can achieve significant speedups relative to the naive method.

From Hard to Soft: Understanding Deep Network Nonlinearities via Vector Quantization and Statistical Inference

Nonlinearity is crucial to the performance of a deep (neural) network (DN). To date there has been little progress understanding the menagerie of available nonlinearities, but recently progress has been made on understanding the r\^ole played by piecewise affine and convex nonlinearities like the ReLU and absolute value activation functions and max-pooling. In particular, DN layers constructed from these operations can be interpreted as {\em max-affine spline operators} (MASOs) that have an elegant link to vector quantization (VQ) and $K$-means. While this is good theoretical progress, the entire MASO approach is predicated on the requirement that the nonlinearities be piecewise affine and convex, which precludes important activation functions like the sigmoid, hyperbolic tangent, and softmax. {\em This paper extends the MASO framework to these and an infinitely large class of new nonlinearities by linking deterministic MASOs with probabilistic Gaussian Mixture Models (GMMs).} We show that, under a GMM, piecewise affine, convex nonlinearities like ReLU, absolute value, and max-pooling can be interpreted as solutions to certain natural "hard" VQ inference problems, while sigmoid, hyperbolic tangent, and softmax can be interpreted as solutions to corresponding "soft" VQ inference problems. We further extend the framework by hybridizing the hard and soft VQ optimizations to create a $\beta$-VQ inference that interpolates between hard, soft, and linear VQ inference. A prime example of a $\beta$-VQ DN nonlinearity is the {\em swish} nonlinearity, which offers state-of-the-art performance in a range of computer vision tasks but was developed ad hoc by experimentation. Finally, we validate with experiments an important assertion of our theory, namely that DN performance can be significantly improved by enforcing orthogonality in its linear filters.

prDeep: Robust Phase Retrieval with a Flexible Deep Network

Phase retrieval algorithms have become an important component in many modern computational imaging systems. For instance, in the context of ptychography and speckle correlation imaging, they enable imaging past the diffraction limit and through scattering media, respectively. Unfortunately, traditional phase retrieval algorithms struggle in the presence of noise. Progress has been made recently on more robust algorithms using signal priors, but at the expense of limiting the range of supported measurement models (e.g., to Gaussian or coded diffraction patterns). In this work we leverage the regularization-by-denoising framework and a convolutional neural network denoiser to create prDeep, a new phase retrieval algorithm that is both robust and broadly applicable. We test and validate prDeep in simulation to demonstrate that it is robust to noise and can handle a variety of system models. A MatConvNet implementation of prDeep is available at https://github.com/ricedsp/prDeep.

MISSION: Ultra Large-Scale Feature Selection using Count-Sketches

Feature selection is an important challenge in machine learning. It plays a crucial role in the explainability of machine-driven decisions that are rapidly permeating throughout modern society. Unfortunately, the explosion in the size and dimensionality of real-world datasets poses a severe challenge to standard feature selection algorithms. Today, it is not uncommon for datasets to have billions of dimensions. At such scale, even storing the feature vector is impossible, causing most existing feature selection methods to fail. Workarounds like feature hashing, a standard approach to large-scale machine learning, helps with the computational feasibility, but at the cost of losing the interpretability of features. In this paper, we present MISSION, a novel framework for ultra large-scale feature selection that performs stochastic gradient descent while maintaining an efficient representation of the features in memory using a Count-Sketch data structure. MISSION retains the simplicity of feature hashing without sacrificing the interpretability of the features while using only O(log^2(p)) working memory. We demonstrate that MISSION accurately and efficiently performs feature selection on real-world, large-scale datasets with billions of dimensions.

Unsupervised Learning with Stein's Unbiased Risk Estimator

Learning from unlabeled and noisy data is one of the grand challenges of machine learning. As such, it has seen a flurry of research with new ideas proposed continuously. In this work, we revisit a classical idea: Stein's Unbiased Risk Estimator (SURE). We show that, in the context of image recovery, SURE and its generalizations can be used to train convolutional neural networks (CNNs) for a range of image denoising and recovery problems {\em without any ground truth data.} Specifically, our goal is to reconstruct an image $x$ from a {\em noisy} linear transformation (measurement) of the image. We consider two scenarios: one where no additional data is available and one where we have measurements of other images that are drawn from the same noisy distribution as $x$, but have no access to the clean images. Such is the case, for instance, in the context of medical imaging, microscopy, and astronomy, where noise-less ground truth data is rarely available. We show that in this situation, SURE can be used to estimate the mean-squared-error loss associated with an estimate of $x$. Using this estimate of the loss, we train networks to perform denoising and compressed sensing recovery. In addition, we also use the SURE framework to partially explain and improve upon an intriguing results presented by Ulyanov et al. in "Deep Image Prior": that a network initialized with random weights and fit to a single noisy image can effectively denoise that image.

Semi-Supervised Learning via New Deep Network Inversion

We exploit a recently derived inversion scheme for arbitrary deep neural networks to develop a new semi-supervised learning framework that applies to a wide range of systems and problems. The approach outperforms current state-of-the-art methods on MNIST reaching $99.14\%$ of test set accuracy while using $5$ labeled examples per class. Experiments with one-dimensional signals highlight the generality of the method. Importantly, our approach is simple, efficient, and requires no change in the deep network architecture.

Learned D-AMP: Principled Neural Network based Compressive Image Recovery

Compressive image recovery is a challenging problem that requires fast and accurate algorithms. Recently, neural networks have been applied to this problem with promising results. By exploiting massively parallel GPU processing architectures and oodles of training data, they can run orders of magnitude faster than existing techniques. However, these methods are largely unprincipled black boxes that are difficult to train and often-times specific to a single measurement matrix. It was recently demonstrated that iterative sparse-signal-recovery algorithms can be "unrolled" to form interpretable deep networks. Taking inspiration from this work, we develop a novel neural network architecture that mimics the behavior of the denoising-based approximate message passing (D-AMP) algorithm. We call this new network Learned D-AMP (LDAMP). The LDAMP network is easy to train, can be applied to a variety of different measurement matrices, and comes with a state-evolution heuristic that accurately predicts its performance. Most importantly, it outperforms the state-of-the-art BM3D-AMP and NLR-CS algorithms in terms of both accuracy and run time. At high resolutions, and when used with sensing matrices that have fast implementations, LDAMP runs over $50\times$ faster than BM3D-AMP and hundreds of times faster than NLR-CS.

DeepCodec: Adaptive Sensing and Recovery via Deep Convolutional Neural Networks

In this paper we develop a novel computational sensing framework for sensing and recovering structured signals. When trained on a set of representative signals, our framework learns to take undersampled measurements and recover signals from them using a deep convolutional neural network. In other words, it learns a transformation from the original signals to a near-optimal number of undersampled measurements and the inverse transformation from measurements to signals. This is in contrast to traditional compressive sensing (CS) systems that use random linear measurements and convex optimization or iterative algorithms for signal recovery. We compare our new framework with $\ell_1$-minimization from the phase transition point of view and demonstrate that it outperforms $\ell_1$-minimization in the regions of phase transition plot where $\ell_1$-minimization cannot recover the exact solution. In addition, we experimentally demonstrate how learning measurements enhances the overall recovery performance, speeds up training of recovery framework, and leads to having fewer parameters to learn.