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UID:941@i2m.univ-amu.fr
DTSTART;TZID=Europe/Paris:20151120T140000
DTEND;TZID=Europe/Paris:20151120T150000
DTSTAMP:20240524T072505Z
URL:https://www.i2m.univ-amu.fr/evenements/emmanuel-soubies-i3s-inria-the-
 continuous-exact-l0-cel0-penalty-an-alternative-to-l0-norm-2/
SUMMARY: (...): Emmanuel Soubies (I3S\, INRIA): The Continuous Exact L0 (CE
 L0) penalty: An alternative to L0-norm
DESCRIPTION:: Title: The Continuous Exact L0 (CEL0) penalty: An alternative
  to L0-normnnAbstract:  Many signal/image processing applications are conc
 erned with sparse estimation/recovery such as compressed sensing\, source 
 separation\, variable separation\, image separation among many others. Usu
 ally\, such a sparsity prior is modeled using the "l0-pseudo norm" countin
 g the nonzero entries of a given vector. This leads to nonconvex optimizat
 ion problems which are well-known to be NP-hard.nAfter an introduction on 
 existing solutions to find a good approximate solution of this problem\, s
 uch that l1 relaxation or greedy algorithms\, we will focus on nonconvex c
 ontinuous penalties approximating the l0-norm for the l0 regularized least
  squares problem. Within this framework\, we will present the Continuous E
 xact l0 penalty (CEL0)\,  an approximation of the l0 norm leading to  a ti
 ght continuous relaxation of the l2-l0 criteria and equal to its convex-hu
 ll when the linear operator\, in the quadratic term\, is orthogonal.  More
 over\, for any linear operator\,  global minimizers of l2-CEL0 contain tho
 se of the l0 penalized least-squares functional. We will also show that fr
 om each local minimizer of this relaxed functional\, one can easily extrac
 t 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 r
 elaxed problem within a macro algorithm ensuring the convergence to a  poi
 nt which is both a critical point of l2-CEL0 and a (local) minimizer of th
 e initial l2-l0 problem.n
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