The linear model, in which a set of observations is assumed to be given by a linear combination of columns of a matrix (often termed a dictionary), has long been the mainstay of the statistics and signal processing li...
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The linear model, in which a set of observations is assumed to be given by a linear combination of columns of a matrix (often termed a dictionary), has long been the mainstay of the statistics and signal processing literature. One particular challenge for inference under linear models is understanding the conditions on the dictionary under which reliable inference is possible. This challenge has attracted renewed attention in recent years, since many modern inference problems (e.g., high-dimensional statistics and compressed sensing) deal with the underdetermined setting, in which the number of observations is much smaller than the number of columns in the dictionary. This paper makes several contributions for this setting when the set of observations is given by a linear combination of a small number of groups of columns of the dictionary, termed the block-sparse case. First, it specifies conditions on the dictionary under which most block submatrices of the dictionary (often termed block subdictionaries) are well conditioned. This result is fundamentally different from prior work on block-sparse inference because: 1) it provides conditions that can be explicitly computed in polynomial time;2) the given conditions translate into near-optimal scaling of the number of columns of the block subdictionaries as a function of the number of observations for a large class of dictionaries;and 3) it suggests that the spectral norm, rather than the column/block coherences of the dictionary, fundamentally limits the scaling of dimensions of the well-conditioned block subdictionaries. Second, in order to help understand the significance of this result in the context of block-sparse inference, this paper investigates the problems of block-sparse recovery and block-sparse regression in underdetermined settings. In both of these problems, this paper utilizes its result concerning conditioning of block subdictionaries and establishes that near-optimal block-sparse recovery and block
In this paper, we survey and compare different algorithms that, given an overcomplete dictionary of elementary functions, solve the problem of simultaneous sparse signal approximation, with common sparsity profile ind...
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In this paper, we survey and compare different algorithms that, given an overcomplete dictionary of elementary functions, solve the problem of simultaneous sparse signal approximation, with common sparsity profile induced by a l(p)-l(q) mixed-norm. Such a problem is also known in the statistical learning community as the group lasso problem. We have gathered and detailed different algorithmic results concerning these two equivalent approximation problems. We have also enriched the discussion by providing relations between several algorithms. Experimental comparisons of the detailed algorithms have also been carried out. The main lesson learned from these experiments is that depending on the performance measure, greedy approaches and iterative reweighted algorithms are the most efficient algorithms either in term of computational complexities, sparsity recovery or mean-square error. (C) 2011 Elsevier B.V. All rights reserved.
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