Sparse approximation

About: Sparse approximation is a research topic. Over the lifetime, 18037 publications have been published within this topic receiving 497739 citations. The topic is also known as: Sparse approximation.

Applications of Sparse Representation and Compressive Sensing [Scanning the Issue]

Abstract: Sparse representation and compressive sensing establishes a more rigorous mathematical framework for studying high-dimensional data and ways to uncover the structures of the data, giving rise to a large repertoire of efficient algorithms. A sparse signal is a signal that can be represented as a linear combination of relatively few base elements in a basis or an overcomplete dictionary. A sufficiently sparse linear representation can be correctly and efficiently computed by greedy methods and convex optimization (i.e., the l1-l0 equivalence), even though this problem is extremely difficult-NP-hard in the general case.

Visual Classification by $\ell _1$ -Hypergraph Modeling

Abstract: Visual classification has attracted considerable research interests in the past decades. In this paper, a novel $\ell _1$ -hypergraph model for visual classification is proposed. Hypergraph learning, as a natural extension of graph model, has been widely used in many machine learning tasks. In previous work, hypergraph is usually constructed by attribute-based or neighborhood-based methods. That is, a hyperedge is generated by connecting a set of samples sharing a same feature attribute or in a neighborhood. However, these methods are unable to explore feature space globally or sensitive to noises. To address these problems, we propose a novel hypergraph construction approach that leverages sparse representation to generate hyperedges and learns the relationship among hyperedges and their vertices. First, for each sample, a hyperedge is generated by regarding it as the centroid and linking it as well as its nearest neighbors. Then, the sparse representation method is applied to represent the centroid vertex by other vertices within the same hyperedge. The vertices with zero coefficients are removed from the hyperedge. Finally, the representation coefficients are used to define the incidence relation between the hyperedge and the vertices. In our approach, we also optimize the hyperedge weights to modulate the effects of different hyperedges. We leverage the prior knowledge on the hyperedges so that the hyperedges sharing more vertices can have closer weights, where a graph Laplacian is used to regularize the optimization of the weights. Our approach is named $\ell _1$ -hypergraph since the $\ell _1$ sparse representation is employed in the hypergraph construction process. The method is evaluated on various visual classification tasks, and it demonstrates promising performance.

Efficient Minimum Error Bounded Particle Resampling L1 Tracker With Occlusion Detection

Abstract: Recently, sparse representation has been applied to visual tracking to find the target with the minimum reconstruction error from a target template subspace. Though effective, these L1 trackers require high computational costs due to numerous calculations for l1 minimization. In addition, the inherent occlusion insensitivity of the l1 minimization has not been fully characterized. In this paper, we propose an efficient L1 tracker, named bounded particle resampling (BPR)-L1 tracker, with a minimum error bound and occlusion detection. First, the minimum error bound is calculated from a linear least squares equation and serves as a guide for particle resampling in a particle filter (PF) framework. Most of the insignificant samples are removed before solving the computationally expensive l1 minimization in a two-step testing. The first step, named τ testing, compares the sample observation likelihood to an ordered set of thresholds to remove insignificant samples without loss of resampling precision. The second step, named max testing, identifies the largest sample probability relative to the target to further remove insignificant samples without altering the tracking result of the current frame. Though sacrificing minimal precision during resampling, max testing achieves significant speed up on top of τ testing. The BPR-L1 technique can also be beneficial to other trackers that have minimum error bounds in a PF framework, especially for trackers based on sparse representations. After the error-bound calculation, BPR-L1 performs occlusion detection by investigating the trivial coefficients in the l1 minimization. These coefficients, by design, contain rich information about image corruptions, including occlusion. Detected occlusions are then used to enhance the template updating. For evaluation, we conduct experiments on three video applications: biometrics (head movement, hand holding object, singers on stage), pedestrians (urban travel, hallway monitoring), and cars in traffic (wide area motion imagery, ground-mounted perspectives). The proposed BPR-L1 method demonstrates an excellent performance as compared with nine state-of-the-art trackers on eleven challenging benchmark sequences.

Sparse Linear Prediction and Its Applications to Speech Processing

Abstract: The aim of this paper is to provide an overview of Sparse Linear Prediction, a set of speech processing tools created by introducing sparsity constraints into the linear prediction framework. These tools have shown to be effective in several issues related to modeling and coding of speech signals. For speech analysis, we provide predictors that are accurate in modeling the speech production process and overcome problems related to traditional linear prediction. In particular, the predictors obtained offer a more effective decoupling of the vocal tract transfer function and its underlying excitation, making it a very efficient method for the analysis of voiced speech. For speech coding, we provide predictors that shape the residual according to the characteristics of the sparse encoding techniques resulting in more straightforward coding strategies. Furthermore, encouraged by the promising application of compressed sensing in signal compression, we investigate its formulation and application to sparse linear predictive coding. The proposed estimators are all solutions to convex optimization problems, which can be solved efficiently and reliably using, e.g., interior-point methods. Extensive experimental results are provided to support the effectiveness of the proposed methods, showing the improvements over traditional linear prediction in both speech analysis and coding.

Working Locally Thinking Globally: Theoretical Guarantees for Convolutional Sparse Coding

Abstract: 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 paper, 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 restricted isometry property 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.