– Emmanuel Soubies (I3S, INRIA) : The Continuous Exact L0 (CEL0) penalty : An alternative to L0-norm

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Date(s) - 20 novembre 2015

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Title : The Continuous Exact L0 (CEL0) penalty : An alternative to L0-norm\n\nAbstract : Many signal/image processing applications are concerned with sparse estimation/recovery such as compressed sensing, source separation, variable separation, image separation among many others. Usually, such a sparsity prior is modeled using the “l0-pseudo norm” counting the nonzero entries of a given vector. This leads to nonconvex optimization problems which are well-known to be NP-hard.\nAfter an introduction on existing solutions to find a good approximate solution of this problem, such that l1 relaxation or greedy algorithms, we will focus on nonconvex continuous penalties approximating the l0-norm for the l0 regularized least squares problem. Within this framework, we will present the Continuous Exact l0 penalty (CEL0), an approximation of the l0 norm leading to a tight continuous relaxation of the l2-l0 criteria and equal to its convex-hull when the linear operator, in the quadratic term, is orthogonal. Moreover, for any linear operator, global minimizers of l2-CEL0 contain those of the l0 penalized least-squares functional. We will also show that from each local minimizer of this relaxed functional, one can easily extract a local minimizer for l2-l0 while the reciprocal is false and some local minimizers of the initial functional are eliminated with l2-CEL0. Hence, the CEL0 functional provides a good continuous alternative to the l2-l0 criteria since it is continuous and convex with respect to each variable. Finally, recent nonsmooth nonconvex algorithms are used to address this relaxed problem within a macro algorithm ensuring the convergence to a point which is both a critical point of l2-CEL0 and a (local) minimizer of the initial l2-l0 problem.\n[