arXiv subjects

Richard G. Baraniuk

Publications and source records attributed to Richard G. Baraniuk.

At least 109 records · Page 6Linked to original sources

arXiv2018

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 rôle 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 $β$-VQ inference that interpolates between hard, soft, and linear VQ inference. A prime example of a $β$-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.

cs.LG↗

arXiv2018

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.

stat.ML↗

arXiv2018

An Expectation-Maximization Approach to Tuning Generalized Vector Approximate Message Passing

Generalized Vector Approximate Message Passing (GVAMP) is an efficient iterative algorithm for approximately minimum-mean-squared-error estimation of a random vector $\mathbf{x}\sim p_{\mathbf{x}}(\mathbf{x})$ from generalized linear measurements, i.e., measurements of the form $\mathbf{y}=Q(\mathbf{z})$ where $\mathbf{z}=\mathbf{Ax}$ with known $\mathbf{A}$, and $Q(\cdot)$ is a noisy, potentially nonlinear, componentwise function. Problems of this form show up in numerous applications, including robust regression, binary classification, quantized compressive sensing, and phase retrieval. In some cases, the prior $p_{\mathbf{x}}$ and/or channel $Q(\cdot)$ depend on unknown deterministic parameters $\boldsymbolθ$, which prevents a direct application of GVAMP. In this paper we propose a way to combine expectation maximization (EM) with GVAMP to jointly estimate $\mathbf{x}$ and $\boldsymbolθ$. We then demonstrate how EM-GVAMP can solve the phase retrieval problem with unknown measurement-noise variance.

cs.IT↗

arXiv2018

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.

cs.DS↗

arXiv2017

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.

stat.ML↗

arXiv2017

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.

stat.ML↗

arXiv2017

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.

stat.ML↗

arXiv2017

Data-Mining Textual Responses to Uncover Misconception Patterns

An important, yet largely unstudied, problem in student data analysis is to detect misconceptions from students' responses to open-response questions. Misconception detection enables instructors to deliver more targeted feedback on the misconceptions exhibited by many students in their class, thus improving the quality of instruction. In this paper, we propose a new natural language processing-based framework to detect the common misconceptions among students' textual responses to short-answer questions. We propose a probabilistic model for students' textual responses involving misconceptions and experimentally validate it on a real-world student-response dataset. Experimental results show that our proposed framework excels at classifying whether a response exhibits one or more misconceptions. More importantly, it can also automatically detect the common misconceptions exhibited across responses from multiple students to multiple questions; this property is especially important at large scale, since instructors will no longer need to manually specify all possible misconceptions that students might exhibit.

stat.ML↗

arXiv2017

Insense: Incoherent Sensor Selection for Sparse Signals

Sensor selection refers to the problem of intelligently selecting a small subset of a collection of available sensors to reduce the sensing cost while preserving signal acquisition performance. The majority of sensor selection algorithms find the subset of sensors that best recovers an arbitrary signal from a number of linear measurements that is larger than the dimension of the signal. In this paper, we develop a new sensor selection algorithm for sparse (or near sparse) signals that finds a subset of sensors that best recovers such signals from a number of measurements that is much smaller than the dimension of the signal. Existing sensor selection algorithms cannot be applied in such situations. Our proposed Incoherent Sensor Selection (Insense) algorithm minimizes a coherence-based cost function that is adapted from recent results in sparse recovery theory. Using six datasets, including two real-world datasets on microbial diagnostics and structural health monitoring, we demonstrate the superior performance of Insense for sparse-signal sensor selection.

cs.OH↗

arXiv2017

Learning to Invert: Signal Recovery via Deep Convolutional Networks

The promise of compressive sensing (CS) has been offset by two significant challenges. First, real-world data is not exactly sparse in a fixed basis. Second, current high-performance recovery algorithms are slow to converge, which limits CS to either non-real-time applications or scenarios where massive back-end computing is available. In this paper, we attack both of these challenges head-on by developing a new signal recovery framework we call {\em DeepInverse} that learns the inverse transformation from measurement vectors to signals using a {\em deep convolutional network}. When trained on a set of representative images, the network learns both a representation for the signals (addressing challenge one) and an inverse map approximating a greedy or convex recovery algorithm (addressing challenge two). Our experiments indicate that the DeepInverse network closely approximates the solution produced by state-of-the-art CS recovery algorithms yet is hundreds of times faster in run time. The tradeoff for the ultrafast run time is a computationally intensive, off-line training procedure typical to deep networks. However, the training needs to be completed only once, which makes the approach attractive for a host of sparse recovery problems.

stat.ML↗

arXiv2016

A Probabilistic Framework for Deep Learning

We develop a probabilistic framework for deep learning based on the Deep Rendering Mixture Model (DRMM), a new generative probabilistic model that explicitly capture variations in data due to latent task nuisance variables. We demonstrate that max-sum inference in the DRMM yields an algorithm that exactly reproduces the operations in deep convolutional neural networks (DCNs), providing a first principles derivation. Our framework provides new insights into the successes and shortcomings of DCNs as well as a principled route to their improvement. DRMM training via the Expectation-Maximization (EM) algorithm is a powerful alternative to DCN back-propagation, and initial training results are promising. Classification based on the DRMM and other variants outperforms DCNs in supervised digit classification, training 2-3x faster while achieving similar accuracy. Moreover, the DRMM is applicable to semi-supervised and unsupervised learning tasks, achieving results that are state-of-the-art in several categories on the MNIST benchmark and comparable to state of the art on the CIFAR10 benchmark.

stat.ML↗

arXiv2016

Semi-Supervised Learning with the Deep Rendering Mixture Model

Semi-supervised learning algorithms reduce the high cost of acquiring labeled training data by using both labeled and unlabeled data during learning. Deep Convolutional Networks (DCNs) have achieved great success in supervised tasks and as such have been widely employed in the semi-supervised learning. In this paper we leverage the recently developed Deep Rendering Mixture Model (DRMM), a probabilistic generative model that models latent nuisance variation, and whose inference algorithm yields DCNs. We develop an EM algorithm for the DRMM to learn from both labeled and unlabeled data. Guided by the theory of the DRMM, we introduce a novel non-negativity constraint and a variational inference term. We report state-of-the-art performance on MNIST and SVHN and competitive results on CIFAR10. We also probe deeper into how a DRMM trained in a semi-supervised setting represents latent nuisance variation using synthetically rendered images. Taken together, our work provides a unified framework for supervised, unsupervised, and semi-supervised learning.

stat.ML↗

arXiv2016

RankMap: A Platform-Aware Framework for Distributed Learning from Dense Datasets

This paper introduces RankMap, a platform-aware end-to-end framework for efficient execution of a broad class of iterative learning algorithms for massive and dense datasets. Our framework exploits data structure to factorize it into an ensemble of lower rank subspaces. The factorization creates sparse low-dimensional representations of the data, a property which is leveraged to devise effective mapping and scheduling of iterative learning algorithms on the distributed computing machines. We provide two APIs, one matrix-based and one graph-based, which facilitate automated adoption of the framework for performing several contemporary learning applications. To demonstrate the utility of RankMap, we solve sparse recovery and power iteration problems on various real-world datasets with up to 1.8 billion non-zeros. Our evaluations are performed on Amazon EC2 and IBM iDataPlex servers using up to 244 cores. The results demonstrate up to two orders of magnitude improvements in memory usage, execution speed, and bandwidth compared with the best reported prior work, while achieving the same level of learning accuracy.

cs.DC↗

arXiv2016

From Denoising to Compressed Sensing

A denoising algorithm seeks to remove noise, errors, or perturbations from a signal. Extensive research has been devoted to this arena over the last several decades, and as a result, today's denoisers can effectively remove large amounts of additive white Gaussian noise. A compressed sensing (CS) reconstruction algorithm seeks to recover a structured signal acquired using a small number of randomized measurements. Typical CS reconstruction algorithms can be cast as iteratively estimating a signal from a perturbed observation. This paper answers a natural question: How can one effectively employ a generic denoiser in a CS reconstruction algorithm? In response, we develop an extension of the approximate message passing (AMP) framework, called Denoising-based AMP (D-AMP), that can integrate a wide class of denoisers within its iterations. We demonstrate that, when used with a high performance denoiser for natural images, D-AMP offers state-of-the-art CS recovery performance while operating tens of times faster than competing methods. We explain the exceptional performance of D-AMP by analyzing some of its theoretical features. A key element in D-AMP is the use of an appropriate Onsager correction term in its iterations, which coerces the signal perturbation at each iteration to be very close to the white Gaussian noise that denoisers are typically designed to remove.

cs.IT↗

arXiv2016

Convex Biclustering

In the biclustering problem, we seek to simultaneously group observations and features. While biclustering has applications in a wide array of domains, ranging from text mining to collaborative filtering, the problem of identifying structure in high dimensional genomic data motivates this work. In this context, biclustering enables us to identify subsets of genes that are co-expressed only within a subset of experimental conditions. We present a convex formulation of the biclustering problem that possesses a unique global minimizer and an iterative algorithm, COBRA, that is guaranteed to identify it. Our approach generates an entire solution path of possible biclusters as a single tuning parameter is varied. We also show how to reduce the problem of selecting this tuning parameter to solving a trivial modification of the convex biclustering problem. The key contributions of our work are its simplicity, interpretability, and algorithmic guarantees - features that arguably are lacking in the current alternative algorithms. We demonstrate the advantages of our approach, which includes stably and reproducibly identifying biclusterings, on simulated and real microarray data.

stat.ME↗

arXiv2016

$k$-POD: A Method for $k$-Means Clustering of Missing Data

The $k$-means algorithm is often used in clustering applications but its usage requires a complete data matrix. Missing data, however, is common in many applications. Mainstream approaches to clustering missing data reduce the missing data problem to a complete data formulation through either deletion or imputation but these solutions may incur significant costs. Our $k$-POD method presents a simple extension of $k$-means clustering for missing data that works even when the missingness mechanism is unknown, when external information is unavailable, and when there is significant missingness in the data.

stat.CO↗

arXiv2015

Consistent Parameter Estimation for LASSO and Approximate Message Passing

We consider the problem of recovering a vector $β_o \in \mathbb{R}^p$ from $n$ random and noisy linear observations $y= Xβ_o + w$, where $X$ is the measurement matrix and $w$ is noise. The LASSO estimate is given by the solution to the optimization problem $\hatβ_λ = \arg \min_β \frac{1}{2} \|y-Xβ\|_2^2 + λ\| β\|_1$. Among the iterative algorithms that have been proposed for solving this optimization problem, approximate message passing (AMP) has attracted attention for its fast convergence. Despite significant progress in the theoretical analysis of the estimates of LASSO and AMP, little is known about their behavior as a function of the regularization parameter $λ$, or the thereshold parameters $τ^t$. For instance the following basic questions have not yet been studied in the literature: (i) How does the size of the active set $\|\hatβ^λ\|_0/p$ behave as a function of $λ$? (ii) How does the mean square error $\|\hatβ_λ - β_o\|_2^2/p$ behave as a function of $λ$? (iii) How does $\|β^t - β_o \|_2^2/p$ behave as a function of $τ^1, \ldots, τ^{t-1}$? Answering these questions will help in addressing practical challenges regarding the optimal tuning of $λ$ or $τ^1, τ^2, \ldots$. This paper answers these questions in the asymptotic setting and shows how these results can be employed in deriving simple and theoretically optimal approaches for tuning the parameters $τ^1, \ldots, τ^t$ for AMP or $λ$ for LASSO. It also explores the connection between the optimal tuning of the parameters of AMP and the optimal tuning of LASSO.

math.ST↗

arXiv2015

Robust 1-Bit Compressive Sensing via Binary Stable Embeddings of Sparse Vectors

The Compressive Sensing (CS) framework aims to ease the burden on analog-to-digital converters (ADCs) by reducing the sampling rate required to acquire and stably recover sparse signals. Practical ADCs not only sample but also quantize each measurement to a finite number of bits; moreover, there is an inverse relationship between the achievable sampling rate and the bit depth. In this paper, we investigate an alternative CS approach that shifts the emphasis from the sampling rate to the number of bits per measurement. In particular, we explore the extreme case of 1-bit CS measurements, which capture just their sign. Our results come in two flavors. First, we consider ideal reconstruction from noiseless 1-bit measurements and provide a lower bound on the best achievable reconstruction error. We also demonstrate that i.i.d. random Gaussian matrices describe measurement mappings achieving, with overwhelming probability, nearly optimal error decay. Next, we consider reconstruction robustness to measurement errors and noise and introduce the Binary $ε$-Stable Embedding (B$ε$SE) property, which characterizes the robustness measurement process to sign changes. We show the same class of matrices that provide almost optimal noiseless performance also enable such a robust mapping. On the practical side, we introduce the Binary Iterative Hard Thresholding (BIHT) algorithm for signal reconstruction from 1-bit measurements that offers state-of-the-art performance.

cs.IT↗