A survey of sparse representation: algorithms and applications

Sparse representation has attracted much attention from researchers in fields of signal processing, image processing, computer vision and pattern recognition. Sparse representation also has a good reputation in both theoretical research and practical applications. Many different algorithms have been proposed for sparse representation. The main purpose of this article is to provide a comprehensive study and an updated review on sparse representation and to supply a guidance for researchers. The taxonomy of sparse representation methods can be studied from various viewpoints. For example, in terms of different norm minimizations used in sparsity constraints, the methods can be roughly categorized into five groups: sparse representation with $l_0$-norm minimization, sparse representation with $l_p$-norm (0$<$p$<$1) minimization, sparse representation with $l_1$-norm minimization and sparse representation with $l_{2,1}$-norm minimization. In this paper, a comprehensive overview of sparse representation is provided. The available sparse representation algorithms can also be empirically categorized into four groups: greedy strategy approximation, constrained optimization, proximity algorithm-based optimization, and homotopy algorithm-based sparse representation. The rationales of different algorithms in each category are analyzed and a wide range of sparse representation applications are summarized, which could sufficiently reveal the potential nature of the sparse representation theory. Specifically, an experimentally comparative study of these sparse representation algorithms was presented. The Matlab code used in this paper can be available at: http://www.yongxu.org/lunwen.html.Comment: Published on IEEE Access, Vol. 3, pp. 490-530, 201

Differentiable Linearized ADMM

Recently, a number of learning-based optimization methods that combine data-driven architectures with the classical optimization algorithms have been proposed and explored, showing superior empirical performance in solving various ill-posed inverse problems, but there is still a scarcity of rigorous analysis about the convergence behaviors of learning-based optimization. In particular, most existing analyses are specific to unconstrained problems but cannot apply to the more general cases where some variables of interest are subject to certain constraints. In this paper, we propose Differentiable Linearized ADMM (D-LADMM) for solving the problems with linear constraints. Specifically, D-LADMM is a K-layer LADMM inspired deep neural network, which is obtained by firstly introducing some learnable weights in the classical Linearized ADMM algorithm and then generalizing the proximal operator to some learnable activation function. Notably, we rigorously prove that there exist a set of learnable parameters for D-LADMM to generate globally converged solutions, and we show that those desired parameters can be attained by training D-LADMM in a proper way. To the best of our knowledge, we are the first to provide the convergence analysis for the learning-based optimization method on constrained problems.Comment: Accepted by ICML201

Optimized Structured Sparse Sensing Matrices for Compressive Sensing

We consider designing a robust structured sparse sensing matrix consisting of a sparse matrix with a few non-zero entries per row and a dense base matrix for capturing signals efficiently We design the robust structured sparse sensing matrix through minimizing the distance between the Gram matrix of the equivalent dictionary and the target Gram of matrix holding small mutual coherence. Moreover, a regularization is added to enforce the robustness of the optimized structured sparse sensing matrix to the sparse representation error (SRE) of signals of interests. An alternating minimization algorithm with global sequence convergence is proposed for solving the corresponding optimization problem. Numerical experiments on synthetic data and natural images show that the obtained structured sensing matrix results in a higher signal reconstruction than a random dense sensing matrix.Comment: 2 tables, 10 figure

A Unified Algorithmic Framework for Block-Structured Optimization Involving Big Data

This article presents a powerful algorithmic framework for big data optimization, called the Block Successive Upper bound Minimization (BSUM). The BSUM includes as special cases many well-known methods for analyzing massive data sets, such as the Block Coordinate Descent (BCD), the Convex-Concave Procedure (CCCP), the Block Coordinate Proximal Gradient (BCPG) method, the Nonnegative Matrix Factorization (NMF), the Expectation Maximization (EM) method and so on. In this article, various features and properties of the BSUM are discussed from the viewpoint of design flexibility, computational efficiency, parallel/distributed implementation and the required communication overhead. Illustrative examples from networking, signal processing and machine learning are presented to demonstrate the practical performance of the BSUM framewor

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

Efficient Sum of Outer Products Dictionary Learning (SOUP-DIL) and Its Application to Inverse Problems

The sparsity of signals in a transform domain or dictionary has been exploited in applications such as compression, denoising and inverse problems. More recently, data-driven adaptation of synthesis dictionaries has shown promise compared to analytical dictionary models. However, dictionary learning problems are typically non-convex and NP-hard, and the usual alternating minimization approaches for these problems are often computationally expensive, with the computations dominated by the NP-hard synthesis sparse coding step. This paper exploits the ideas that drive algorithms such as K-SVD, and investigates in detail efficient methods for aggregate sparsity penalized dictionary learning by first approximating the data with a sum of sparse rank-one matrices (outer products) and then using a block coordinate descent approach to estimate the unknowns. The resulting block coordinate descent algorithms involve efficient closed-form solutions. Furthermore, we consider the problem of dictionary-blind image reconstruction, and propose novel and efficient algorithms for adaptive image reconstruction using block coordinate descent and sum of outer products methodologies. We provide a convergence study of the algorithms for dictionary learning and dictionary-blind image reconstruction. Our numerical experiments show the promising performance and speed-ups provided by the proposed methods over previous schemes in sparse data representation and compressed sensing-based image reconstruction.Comment: Accepted to IEEE Transactions on Computational Imaging. This paper also cites experimental results reported in arXiv:1511.0884

Learning a collaborative multiscale dictionary based on robust empirical mode decomposition

Dictionary learning is a challenge topic in many image processing areas. The basic goal is to learn a sparse representation from an overcomplete basis set. Due to combining the advantages of generic multiscale representations with learning based adaptivity, multiscale dictionary representation approaches have the power in capturing structural characteristics of natural images. However, existing multiscale learning approaches still suffer from three main weaknesses: inadaptability to diverse scales of image data, sensitivity to noise and outliers, difficulty to determine optimal dictionary structure. In this paper, we present a novel multiscale dictionary learning paradigm for sparse image representations based on an improved empirical mode decomposition. This powerful data-driven analysis tool for multi-dimensional signal can fully adaptively decompose the image into multiscale oscillating components according to intrinsic modes of data self. This treatment can obtain a robust and effective sparse representation, and meanwhile generates a raw base dictionary at multiple geometric scales and spatial frequency bands. This dictionary is refined by selecting optimal oscillating atoms based on frequency clustering. In order to further enhance sparsity and generalization, a tolerance dictionary is learned using a coherence regularized model. A fast proximal scheme is developed to optimize this model. The multiscale dictionary is considered as the product of oscillating dictionary and tolerance dictionary. Experimental results demonstrate that the proposed learning approach has the superior performance in sparse image representations as compared with several competing methods. We also show the promising results in image denoising application.Comment: to be published in Neurocomputin

ADMM for Multiaffine Constrained Optimization

We expand the scope of the alternating direction method of multipliers (ADMM). Specifically, we show that ADMM, when employed to solve problems with multiaffine constraints that satisfy certain verifiable assumptions, converges to the set of constrained stationary points if the penalty parameter in the augmented Lagrangian is sufficiently large. When the Kurdyka-\L{}ojasiewicz (K-\L{}) property holds, this is strengthened to convergence to a single constrained stationary point. Our analysis applies under assumptions that we have endeavored to make as weak as possible. It applies to problems that involve nonconvex and/or nonsmooth objective terms, in addition to the multiaffine constraints that can involve multiple (three or more) blocks of variables. To illustrate the applicability of our results, we describe examples including nonnegative matrix factorization, sparse learning, risk parity portfolio selection, nonconvex formulations of convex problems, and neural network training. In each case, our ADMM approach encounters only subproblems that have closed-form solutions.Comment: v3: 37 pages, 7 figures v2: 32 pages, 0 figures. v1: 26 pages, 0 figure

Proximal methods for the latent group lasso penalty

We consider a regularized least squares problem, with regularization by structured sparsity-inducing norms, which extend the usual $\ell_1$ and the group lasso penalty, by allowing the subsets to overlap. Such regularizations lead to nonsmooth problems that are difficult to optimize, and we propose in this paper a suitable version of an accelerated proximal method to solve them. We prove convergence of a nested procedure, obtained composing an accelerated proximal method with an inner algorithm for computing the proximity operator. By exploiting the geometrical properties of the penalty, we devise a new active set strategy, thanks to which the inner iteration is relatively fast, thus guaranteeing good computational performances of the overall algorithm. Our approach allows to deal with high dimensional problems without pre-processing for dimensionality reduction, leading to better computational and prediction performances with respect to the state-of-the art methods, as shown empirically both on toy and real data.Comment: 4 figure

Decomposition into Low-rank plus Additive Matrices for Background/Foreground Separation: A Review for a Comparative Evaluation with a Large-Scale Dataset

Recent research on problem formulations based on decomposition into low-rank plus sparse matrices shows a suitable framework to separate moving objects from the background. The most representative problem formulation is the Robust Principal Component Analysis (RPCA) solved via Principal Component Pursuit (PCP) which decomposes a data matrix in a low-rank matrix and a sparse matrix. However, similar robust implicit or explicit decompositions can be made in the following problem formulations: Robust Non-negative Matrix Factorization (RNMF), Robust Matrix Completion (RMC), Robust Subspace Recovery (RSR), Robust Subspace Tracking (RST) and Robust Low-Rank Minimization (RLRM). The main goal of these similar problem formulations is to obtain explicitly or implicitly a decomposition into low-rank matrix plus additive matrices. In this context, this work aims to initiate a rigorous and comprehensive review of the similar problem formulations in robust subspace learning and tracking based on decomposition into low-rank plus additive matrices for testing and ranking existing algorithms for background/foreground separation. For this, we first provide a preliminary review of the recent developments in the different problem formulations which allows us to define a unified view that we called Decomposition into Low-rank plus Additive Matrices (DLAM). Then, we examine carefully each method in each robust subspace learning/tracking frameworks with their decomposition, their loss functions, their optimization problem and their solvers. Furthermore, we investigate if incremental algorithms and real-time implementations can be achieved for background/foreground separation. Finally, experimental results on a large-scale dataset called Background Models Challenge (BMC 2012) show the comparative performance of 32 different robust subspace learning/tracking methods.Comment: 121 pages, 5 figures, submitted to Computer Science Review. arXiv admin note: text overlap with arXiv:1312.7167, arXiv:1109.6297, arXiv:1207.3438, arXiv:1105.2126, arXiv:1404.7592, arXiv:1210.0805, arXiv:1403.8067 by other authors, Computer Science Review, November 201