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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Sensitivity Analysis with Degeneracy: Mirror Strat
ifiable Functions - Jalal Fadili (None / Other)
DTSTART;TZID=Europe/London:20170906T140000
DTEND;TZID=Europe/London:20170906T145000
UID:TALK78101AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/78101
DESCRIPTION:This talk will present a set of sensitivity analys
is and activity identification results for a class
of convex functions with a strong geometric struc
ture\, that we coin ``mirror-stratifiable'\;
9\;. These functions are such that there is a bije
ction between a primal and a dual stratification o
f the space into partitioning sets\, called strata
. This pairing is crucial to track the strata that
are identifiable by solutions of parametrized opt
imization problems or by iterates of optimization
algorithms. This class of functions encompasses al
l regularizers routinely used in signal and image
processing\, machine learning\, and statistics. We
show that this ``mirror-stratifiable'\;'\;
structure enjoys a nice sensitivity theory\, allow
ing us to study stability of solutions of optimiza
tion problems to small perturbations\, as well as
activity identification of first-order proximal sp
litting-type algorithms.

Existing results
in the literature typically assume that\, under a
non-degeneracy condition\, the active set associat
ed to a minimizer is stable to small perturbations
and is identified in finite time by optimization
schemes. In contrast\, our results do not require
any non-degeneracy assumption: in consequence\, th
e optimal active set is not necessarily stable any
more\, but we are able to track precisely the set
of identifiable strata. We show that these results
have crucial implications when solving challengin
g ill-posed inverse problems via regularization\,
a typical scenario where the non-degeneracy condit
ion is not fulfilled. Our theoretical results\, il
lustrated by numerical simulations\, \; allow
to characterize the instability behaviour of the r
egularized solutions\, by locating the set of all
low-dimensional strata that can be potentially ide
ntified by these solutions.

This is a joint
work with Jé\;rô\;me Malick and Gabrie
l Peyré\;.

LOCATION:Seminar Room 1\, Newton Institute
CONTACT:INI IT
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