Abstract. Sparse representation of signals has been the focus of much research in the recent years. A vast majority of existing algorithms deal with vectors, and higher–order data like images are dealt with by vectorization. However, the structure of the data may be lost in the process, leading to a poorer representation and overall performance degradation. In this paper we propose a novel approach for sparse representation of positive definite matrices, where vectorization will destroy the inherent structure of the data. The sparse decomposition of a positive definite matrix is formulated as a convex optimization problem, which falls under the category of determinant maximization (MAXDET) problems [1], for which efficient interior point algorithms exist. Experimental results are shown with simulated examples as well as in real–world computer vision applications, demonstrating the suitability of the new model. This forms the first step toward extending the cornucopia of sparsity-based algorithms to positive definite matrices.

Abstract—This letter proposes a simultaneous joint sparsity model for target detection in hyperspectral imagery (HSI). The key innovative idea here is that hyperspectral pixels within a small neighborhood in the test image can be simultaneously represented by a linear combination of a few common training samples but weighted with a different set of coefficients for each pixel. The joint sparsity model automatically incorporates the interpixel correlation within the HSI by assuming that neighboring pixels usually consist of similar materials. The sparse representations of the neighboring pixels are obtained by simultaneously decomposing the pixels over a given dictionary consisting of training samples of both the target and background classes. The recovered sparse coefficient vectors are then directly used for determining the label of the test pixels. Simulation results show that the proposed algorithm outperforms the classical hyperspectral target detection algorithms, such as the popular spectral matched filters, matched subspace detectors, and adaptive subspace detectors, as well as binary classifiers such as support vector machines. Index Terms—Hyperspectral imagery, joint sparsity model, simultaneous orthogonal matching pursuit, sparse representation, target detection. I.

Abstract. The conditional correlation patterns of an anatomical shape may provide some important information on the structure of this shape. We propose to investigate these patterns by Gaussian Graphical Modelling. We design a model which takes into account both local and longdistance dependencies. We provide an algorithm which estimates sparse long-distance conditional correlations, highlighting the most significant ones. The selection procedure is based on a criterion which quantifies the quality of the conditional correlation graph in terms of prediction. The preliminary results on AD versus control population show noticeable differences.

In this paper, we propose a novel multi-task multivariate (MTMV) sparse representation method for multi-sensor classification, which takes into account correlations as well as complementary information between heterogeneous sensors simultaneously while considering joint sparsity within each sensor’s observations. This approach can be seen as the generalized model of multi-task and multivariate Lasso, where all the multi-sensor data are jointly represented by a sparse linear combination of the training data. We further modify our MTMV model by including an environmental clutter noise term that is also assumed to be sparse in the feature domain. An efficient algorithm based on alternative direction method is proposed for both models. Extensive experiments are conducted on real data sets and the results are compared with the conventional discriminative classifiers to verify the effectiveness of the proposed methods in the application of automatic border patrol, where we often have to discriminate between human and animal footsteps. I.

Abstract—In this paper, a novel nonlinear technique for hyperspectral image classification is proposed. Our approach relies on sparsely representing a test sample in terms of all of the training samples in a feature space induced by a kernel function. For each test pixel in the feature space, a sparse representation vector is obtained by decomposing the test pixel over a training dictionary, also in the same feature space, by using a kernel-based greedy pursuit algorithm. The recovered sparse representation vector is then used directly to determine the class label of the test pixel. Projecting the samples into a high-dimensional feature space and kernelizing the sparse representation improves the data separability between different classes, providing a higher classification accuracy compared to the more conventional linear sparsity-based classification algorithms. Moreover, the spatial coherency across neighboring pixels is also incorporated through a kernelized joint sparsity model, where all of the pixels within a small neighborhood are jointly represented in the feature space by selecting a few common training samples. Kernel greedy optimization algorithms are suggested in this paper to solve the kernel versions of the single-pixel and multi-pixel joint sparsity-based recovery problems. Experimental results on several hyperspectral images show that the proposed technique outperforms the linear sparsity-based classification technique, as well as the classical Support Vector Machines and sparse kernel logistic regression classifiers. I.

recoverability from dense corrupted

(will be inserted by the editor) Bounds of restricted isometry constants in extreme asymptotics: formulae for Gaussian matrices Bubacarr Bah · Jared Tanner the date of receipt and acceptance should be inserted later Abstract Restricted Isometry Constants (RICs) provide a measure of how far from an isometry a matrix can be when acting on sparse vectors. This, and related quantities, provide a mechanism by which standard eigen-analysis can be applied to topics relying on sparsity. RIC bounds have been presented for a variety of random matrices and matrix dimension and sparsity ranges. We provide explicitly formulae for RIC bounds, of n ×N Gaussian matrices with sparsity k, in three settings: a) n/N fixed and k/n approaching zero, b) k/n fixed and n/N approaching zero, and c) n/N approaching zero with k/n decaying inverse logrithmically in N/n; in these three settings the RICs a) decay to zero, b) become unbounded (or approach inherent bounds), and c) approach a non-zero constant. Implications of these results for RIC based analysis of compressed sensing algorithms are presented.

In this paper, we consider the problem of compressed sensing where the goal is to recover almost all sparse vectors using a small number of fixed linear measurements. For this problem, we propose a novel partial hard-thresholding operator that leads to a general family of iterative algorithms. While one extreme of the family yields well known hard thresholding algorithms like ITI and HTP[17, 10], the other end of the spectrum leads to a novel algorithm that we call Orthogonal Matching Pursuit with Replacement (OMPR). OMPR, like the classic greedy algorithm OMP, adds exactly one coordinate to the support at each iteration, based on the correlation with the current residual. However, unlike OMP, OMPR also removes one coordinate from the support. This simple change allows us to prove that OMPR has the best known guarantees for sparse recovery in terms of the Restricted Isometry Property (a condition on the measurement matrix). In contrast, OMP is known to have very weak performance guarantees under RIP. Given its simple structure, we are able to extend OMPR using locality sensitive hashing to get OMPR-Hash, the first provably sub-linear (in dimensionality) algorithm for sparse recovery. Our proof techniques are novel and flexible enough to also permit the tightest known analysis of popular iterative algorithms such as CoSaMP and Subspace Pursuit. We provide experimental results on large problems providing recovery for vectors of size up to million dimensions. We demonstrate that for large-scale problems our proposed methods are more robust and faster than existing methods. 1

Learning sparse representations on data adaptive dictionaries is a state-of-the-art method for modeling data. But when the dictionary is large and the data dimension is high, it is a computationally challenging problem. We explore three aspects of the problem. First, we derive new, greatly improved screening tests that quickly identify codewords that are guaranteed to have zero weights. Second, we study the properties of random projections in the context of learning sparse representations. Finally, we develop a hierarchical framework that uses incremental random projections and screening to learn, in small stages, a hierarchically structured dictionary for sparse representations. Empirical results show that our framework can learn informative hierarchical sparse representations more efficiently. 1

Many machine learning and signal processing problems can be formulated as linearly constrained convex programs, which could be efficiently solved by the alternating direction method (ADM). However, usually the subproblems in ADM are easily solvable only when the linear mappings in the constraints are identities. To address this issue, we propose a linearized ADM (LADM) method by linearizing the quadratic penalty term and adding a proximal term when solving the subproblems. For fast convergence, we also allow the penalty to change adaptively according a novel update rule. We prove the global convergence of LADM with adaptive penalty (LADMAP). As an example, we apply LADMAP to solve lowrank representation (LRR), which is an important subspace clustering technique yet suffers from high computation cost. By combining LADMAP with a skinny SVD representation technique, we are able to reduce the complexity O(n 3) of the original ADM based method to O(rn 2), where r and n are the rank and size of the representation matrix, respectively, hence making LRR possible for large scale applications. Numerical experiments verify that for LRR our LADMAP based methods are much faster than state-of-the-art algorithms. 1