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

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Date(s) - 20/11/2015
14 h 00 min - 15 h 00 min

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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

Olivier CHABROL
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