arXiv subjects

Jeremias Sulam

Publications and source records attributed to Jeremias Sulam.

48 records · Page 3Linked to original sources

arXiv2019

Adversarial Noise Attacks of Deep Learning Architectures -- Stability Analysis via Sparse Modeled Signals

Despite their impressive performance, deep convolutional neural networks (CNNs) have been shown to be sensitive to small adversarial perturbations. These nuisances, which one can barely notice, are powerful enough to fool sophisticated and well performing classifiers, leading to ridiculous misclassification results. In this paper we analyze the stability of state-of-the-art deep-learning classification machines to adversarial perturbations, where we assume that the signals belong to the (possibly multi-layer) sparse representation model. We start with convolutional sparsity and then proceed to its multi-layered version, which is tightly connected to CNNs. Our analysis links between the stability of the classification to noise and the underlying structure of the signal, quantified by the sparsity of its representation under a fixed dictionary. In addition, we offer similar stability theorems for two practical pursuit algorithms, which are posed as two different deep-learning architectures - the layered Thresholding and the layered Basis Pursuit. Our analysis establishes the better robustness of the later to adversarial attacks. We corroborate these theoretical results by numerical experiments on three datasets: MNIST, CIFAR-10 and CIFAR-100.

stat.ML↗

arXiv2019

MMSE Approximation For Sparse Coding Algorithms Using Stochastic Resonance

Sparse coding refers to the pursuit of the sparsest representation of a signal in a typically overcomplete dictionary. From a Bayesian perspective, sparse coding provides a Maximum a Posteriori (MAP) estimate of the unknown vector under a sparse prior. In this work, we suggest enhancing the performance of sparse coding algorithms by a deliberate and controlled contamination of the input with random noise, a phenomenon known as stochastic resonance. The proposed method adds controlled noise to the input and estimates a sparse representation from the perturbed signal. A set of such solutions is then obtained by projecting the original input signal onto the recovered set of supports. We present two variants of the described method, which differ in their final step. The first is a provably convergent approximation to the Minimum Mean Square Error (MMSE) estimator, relying on the generative model and applying a weighted average over the recovered solutions. The second is a relaxed variant of the former that simply applies an empirical mean. We show that both methods provide a computationally efficient approximation to the MMSE estimator, which is typically intractable to compute. We demonstrate our findings empirically and provide a theoretical analysis of our method under several different cases.

eess.SP↗

arXiv2018

On Multi-Layer Basis Pursuit, Efficient Algorithms and Convolutional Neural Networks

Parsimonious representations are ubiquitous in modeling and processing information. Motivated by the recent Multi-Layer Convolutional Sparse Coding (ML-CSC) model, we herein generalize the traditional Basis Pursuit problem to a multi-layer setting, introducing similar sparse enforcing penalties at different representation layers in a symbiotic relation between synthesis and analysis sparse priors. We explore different iterative methods to solve this new problem in practice, and we propose a new Multi-Layer Iterative Soft Thresholding Algorithm (ML-ISTA), as well as a fast version (ML-FISTA). We show that these nested first order algorithms converge, in the sense that the function value of near-fixed points can get arbitrarily close to the solution of the original problem. We further show how these algorithms effectively implement particular recurrent convolutional neural networks (CNNs) that generalize feed-forward ones without introducing any parameters. We present and analyze different architectures resulting unfolding the iterations of the proposed pursuit algorithms, including a new Learned ML-ISTA, providing a principled way to construct deep recurrent CNNs. Unlike other similar constructions, these architectures unfold a global pursuit holistically for the entire network. We demonstrate the emerging constructions in a supervised learning setting, consistently improving the performance of classical CNNs while maintaining the number of parameters constant.

cs.LG↗

arXiv2018

A Local Block Coordinate Descent Algorithm for the Convolutional Sparse Coding Model

The Convolutional Sparse Coding (CSC) model has recently gained considerable traction in the signal and image processing communities. By providing a global, yet tractable, model that operates on the whole image, the CSC was shown to overcome several limitations of the patch-based sparse model while achieving superior performance in various applications. Contemporary methods for pursuit and learning the CSC dictionary often rely on the Alternating Direction Method of Multipliers (ADMM) in the Fourier domain for the computational convenience of convolutions, while ignoring the local characterizations of the image. A recent work by Papyan et al. suggested the SBDL algorithm for the CSC, while operating locally on image patches. SBDL demonstrates better performance compared to the Fourier-based methods, albeit still relying on the ADMM. In this work we maintain the localized strategy of the SBDL, while proposing a new and much simpler approach based on the Block Coordinate Descent algorithm - this method is termed Local Block Coordinate Descent (LoBCoD). Furthermore, we introduce a novel stochastic gradient descent version of LoBCoD for training the convolutional filters. The Stochastic-LoBCoD leverages the benefits of online learning, while being applicable to a single training image. We demonstrate the advantages of the proposed algorithms for image inpainting and multi-focus image fusion, achieving state-of-the-art results.

cs.CV↗

arXiv2018

Variations on the CSC model

Over the past decade, the celebrated sparse representation model has achieved impressive results in various signal and image processing tasks. A convolutional version of this model, termed convolutional sparse coding (CSC), has been recently reintroduced and extensively studied. CSC brings a natural remedy to the limitation of typical sparse enforcing approaches of handling global and high-dimensional signals by local, patch-based, processing. While the classic field of sparse representations has been able to cater for the diverse challenges of different signal processing tasks by considering a wide range of problem formulations, almost all available algorithms that deploy the CSC model consider the same $\ell_1 - \ell_2$ problem form. As we argue in this paper, this CSC pursuit formulation is also too restrictive as it fails to explicitly exploit some local characteristics of the signal. This work expands the range of formulations for the CSC model by proposing two convex alternatives that merge global norms with local penalties and constraints. The main contribution of this work is the derivation of efficient and provably converging algorithms to solve these new sparse coding formulations.

eess.SP↗

arXiv2018

Multi-Layer Sparse Coding: The Holistic Way

The recently proposed multi-layer sparse model has raised insightful connections between sparse representations and convolutional neural networks (CNN). In its original conception, this model was restricted to a cascade of convolutional synthesis representations. In this paper, we start by addressing a more general model, revealing interesting ties to fully connected networks. We then show that this multi-layer construction admits a brand new interpretation in a unique symbiosis between synthesis and analysis models: while the deepest layer indeed provides a synthesis representation, the mid-layers decompositions provide an analysis counterpart. This new perspective exposes the suboptimality of previously proposed pursuit approaches, as they do not fully leverage all the information comprised in the model constraints. Armed with this understanding, we address fundamental theoretical issues, revisiting previous analysis and expanding it. Motivated by the limitations of previous algorithms, we then propose an integrated - holistic - alternative that estimates all representations in the model simultaneously, and analyze all these different schemes under stochastic noise assumptions. Inspired by the synthesis-analysis duality, we further present a Holistic Pursuit algorithm, which alternates between synthesis and analysis sparse coding steps, eventually solving for the entire model as a whole, with provable improved performance. Finally, we present numerical results that demonstrate the practical advantages of our approach.

eess.IV↗

arXiv2018

Multi-Layer Convolutional Sparse Modeling: Pursuit and Dictionary Learning

The recently proposed Multi-Layer Convolutional Sparse Coding (ML-CSC) model, consisting of a cascade of convolutional sparse layers, provides a new interpretation of Convolutional Neural Networks (CNNs). Under this framework, the computation of the forward pass in a CNN is equivalent to a pursuit algorithm aiming to estimate the nested sparse representation vectors -- or feature maps -- from a given input signal. Despite having served as a pivotal connection between CNNs and sparse modeling, a deeper understanding of the ML-CSC is still lacking: there are no pursuit algorithms that can serve this model exactly, nor are there conditions to guarantee a non-empty model. While one can easily obtain signals that approximately satisfy the ML-CSC constraints, it remains unclear how to simply sample from the model and, more importantly, how one can train the convolutional filters from real data. In this work, we propose a sound pursuit algorithm for the ML-CSC model by adopting a projection approach. We provide new and improved bounds on the stability of the solution of such pursuit and we analyze different practical alternatives to implement this in practice. We show that the training of the filters is essential to allow for non-trivial signals in the model, and we derive an online algorithm to learn the dictionaries from real data, effectively resulting in cascaded sparse convolutional layers. Last, but not least, we demonstrate the applicability of the ML-CSC model for several applications in an unsupervised setting, providing competitive results. Our work represents a bridge between matrix factorization, sparse dictionary learning and sparse auto-encoders, and we analyze these connections in detail.

cs.CV↗

arXiv2017

Working Locally Thinking Globally: Theoretical Guarantees for Convolutional Sparse Coding

The celebrated sparse representation model has led to remarkable results in various signal processing tasks in the last decade. However, despite its initial purpose of serving as a global prior for entire signals, it has been commonly used for modeling low dimensional patches due to the computational constraints it entails when deployed with learned dictionaries. A way around this problem has been recently proposed, adopting a convolutional sparse representation model. This approach assumes that the global dictionary is a concatenation of banded Circulant matrices. While several works have presented algorithmic solutions to the global pursuit problem under this new model, very few truly-effective guarantees are known for the success of such methods. In this work, we address the theoretical aspects of the convolutional sparse model providing the first meaningful answers to questions of uniqueness of solutions and success of pursuit algorithms, both greedy and convex relaxations, in ideal and noisy regimes. To this end, we generalize mathematical quantities, such as the $\ell_0$ norm, mutual coherence, Spark and RIP to their counterparts in the convolutional setting, intrinsically capturing local measures of the global model. On the algorithmic side, we demonstrate how to solve the global pursuit problem by using simple local processing, thus offering a first of its kind bridge between global modeling of signals and their patch-based local treatment.

cs.IT↗

arXiv2017

Convolutional Dictionary Learning via Local Processing

Convolutional Sparse Coding (CSC) is an increasingly popular model in the signal and image processing communities, tackling some of the limitations of traditional patch-based sparse representations. Although several works have addressed the dictionary learning problem under this model, these relied on an ADMM formulation in the Fourier domain, losing the sense of locality and the relation to the traditional patch-based sparse pursuit. A recent work suggested a novel theoretical analysis of this global model, providing guarantees that rely on a localized sparsity measure. Herein, we extend this local-global relation by showing how one can efficiently solve the convolutional sparse pursuit problem and train the filters involved, while operating locally on image patches. Our approach provides an intuitive algorithm that can leverage standard techniques from the sparse representations field. The proposed method is fast to train, simple to implement, and flexible enough that it can be easily deployed in a variety of applications. We demonstrate the proposed training scheme for image inpainting and image separation, while achieving state-of-the-art results.

cs.CV↗

arXiv2017

Working Locally Thinking Globally - Part I: Theoretical Guarantees for Convolutional Sparse Coding

The celebrated sparse representation model has led to remarkable results in various signal processing tasks in the last decade. However, despite its initial purpose of serving as a global prior for entire signals, it has been commonly used for modeling low dimensional patches due to the computational constraints it entails when deployed with learned dictionaries. A way around this problem has been proposed recently, adopting a convolutional sparse representation model. This approach assumes that the global dictionary is a concatenation of banded Circulant matrices. Although several works have presented algorithmic solutions to the global pursuit problem under this new model, very few truly-effective guarantees are known for the success of such methods. In the first of this two-part work, we address the theoretical aspects of the sparse convolutional model, providing the first meaningful answers to corresponding questions of uniqueness of solutions and success of pursuit algorithms. To this end, we generalize mathematical quantities, such as the $\ell_0$ norm, the mutual coherence and the Spark, to their counterparts in the convolutional setting, which intrinsically capture local measures of the global model. In a companion paper, we extend the analysis to a noisy regime, addressing the stability of the sparsest solutions and pursuit algorithms, and demonstrate practical approaches for solving the global pursuit problem via simple local processing.

cs.IT↗

arXiv2017

Working Locally Thinking Globally - Part II: Stability and Algorithms for Convolutional Sparse Coding

The convolutional sparse model has recently gained increasing attention in the signal and image processing communities, and several methods have been proposed for solving the pursuit problem emerging from it -- in particular its convex relaxation, Basis Pursuit. In the first of this two-part work, we have provided a theoretical back-bone for this model, providing guarantees for the uniqueness of the sparsest solution and for the success of pursuit algorithms by introducing the notion of stripe sparsity and other related measures. Herein, we extend the analysis to a noisy regime, thereby considering signal perturbations and model deviations. We address questions of stability of the sparsest solutions and the success of pursuit algorithms, both greedy and convex. Classical definitions such as the RIP are generalized to the convolutional model, and existing notions such as the ERC are connected to our setting. On the algorithmic side, we demonstrate how to solve the global pursuit problem by using simple local processing, thus offering a first of its kind bridge between global modeling of signals and their patch-based local treatment.

cs.IT↗

arXiv2016

Trainlets: Dictionary Learning in High Dimensions

Sparse representations has shown to be a very powerful model for real world signals, and has enabled the development of applications with notable performance. Combined with the ability to learn a dictionary from signal examples, sparsity-inspired algorithms are often achieving state-of-the-art results in a wide variety of tasks. Yet, these methods have traditionally been restricted to small dimensions mainly due to the computational constraints that the dictionary learning problem entails. In the context of image processing, this implies handling small image patches. In this work we show how to efficiently handle bigger dimensions and go beyond the small patches in sparsity-based signal and image processing methods. We build our approach based on a new cropped wavelet decomposition, which enables a multi-scale analysis with virtually no border effects. We then employ this as the base dictionary within a double sparsity model to enable the training of adaptive dictionaries. To cope with the increase of training data, while at the same time improving the training performance, we present an Online Sparse Dictionary Learning (OSDL) algorithm to train this model effectively, enabling it to handle millions of examples. This work shows that dictionary learning can be up-scaled to tackle a new level of signal dimensions, obtaining large adaptable atoms that we call trainlets.

cs.CV↗