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SUMMARY:Sensitivity Analysis with Degeneracy: Mirror Stratifiable Function
 s - Jalal Fadili (None / Other)
DTSTART:20170906T130000Z
DTEND:20170906T135000Z
UID:TALK78101@talks.cam.ac.uk
CONTACT:INI IT
DESCRIPTION:This talk will present a set of sensitivity analysis and activ
 ity identification results for a class of convex functions with a strong g
 eometric structure\, that we coin ``mirror-stratifiable&#39\;&#39\;. These
  functions are such that there is a bijection between a primal and a dual 
 stratification of the space into partitioning sets\, called strata. This p
 airing is crucial to track the strata that are identifiable by solutions o
 f parametrized optimization problems or by iterates of optimization algori
 thms. This class of functions encompasses all regularizers routinely used 
 in signal and image processing\, machine learning\, and statistics. We sho
 w that this ``mirror-stratifiable&#39\;&#39\; structure enjoys a nice sens
 itivity theory\, allowing us to study stability of solutions of optimizati
 on problems to small perturbations\, as well as activity identification of
  first-order proximal splitting-type algorithms. <br><br>Existing results 
 in the literature typically assume that\, under a non-degeneracy condition
 \, the active set associated to a minimizer is stable to small perturbatio
 ns and is identified in finite time by optimization schemes. In contrast\,
  our results do not require any non-degeneracy assumption: in consequence\
 , the optimal active set is not necessarily stable anymore\, but we are ab
 le to track precisely the set of identifiable strata. We show that these r
 esults have crucial implications when solving challenging ill-posed invers
 e problems via regularization\, a typical scenario where the non-degenerac
 y condition is not fulfilled. Our theoretical results\, illustrated by num
 erical simulations\,&nbsp\; allow to characterize the instability behaviou
 r of the regularized solutions\, by locating the set of all low-dimensiona
 l strata that can be potentially identified by these solutions.<br><br>Thi
 s is a joint work with J&eacute\;r&ocirc\;me Malick and Gabriel Peyr&eacut
 e\;.<br><br>
LOCATION:Seminar Room 1\, Newton Institute
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