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 HAL : hal-00260707, version 1
 arXiv : 0803.0524
 Numerical Functional Analysis and Optimization 32, 7 (2011) 768-805
 Average performance of the sparsest approximation using a general dictionary
 (2011)
 Let $A$ be a matrix of size $M\times N$ (a dictionary) and let $\|.\|$ be a norm on $\RR^N$. For any $d\in\RR^N$ we consider the sparsest vector (i.e. the one with the smallest number of non zero entries) $u\in\RR^M$ such that $\|Au-d\| \leq \tau$, for a parameter $\tau>0$. We say that $u$ is a $K$-sparse solution if it has less than $K\in\NN$ non zero entries. In this paper, we give a precise geometrical description of the data sets yielding a $K$-sparse solution. We parameterize and measure these sets. More precisely, we measure their intersection with a ball defined by any given norm $\delta$ and a radius $\theta$. These measures are expressed in terms of the constituents of the optimization problem. This is the core of a new methodology, called Average Performance in Approximation (APA), inaugurated in this work. By way of application, we give the probability of obtaining a $K$-sparse solution, when $d$ is uniformly distributed in the $\delta$-ball of radius $\theta$. Analyzing the obtained formulas reveals what are the most important features of the dictionary and the norm defining the data fidelity, to obtain sparse solutions. This important question is largely discussed. We also provide an example when both $\|.\|$ and $\delta$ are the Euclidian norm. Some among the wide-ranging perspectives raised by the new APA methodology are described as well.
 1 : Laboratoire Analyse, Géométrie et Application (LAGA) CNRS : UMR7539 – Université Paris XIII - Paris Nord – Université Paris VIII - Vincennes Saint-Denis 2 : Centre de Mathématiques et de Leurs Applications (CMLA) CNRS : UMR8536 – École normale supérieure de Cachan - ENS Cachan
 Domaine : Mathématiques/Optimisation et contrôle
 Mots Clés : compression – approximation – best K-term approximation – constrained minimization – dictionary – $\ell_0$ norm – estimation – frames – measure theory – nonconvex functions – sparse representations.
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 hal-00260707, version 1 http://hal.archives-ouvertes.fr/hal-00260707 oai:hal.archives-ouvertes.fr:hal-00260707 Contributeur : Francois Malgouyres <> Soumis le : Mardi 4 Mars 2008, 17:52:26 Dernière modification le : Vendredi 4 Novembre 2011, 16:00:44