%0 Journal Article
%T Exact Sparse Approximation Problems via Mixed-Integer Programming: Formulations and Computational Performance
%+ Institut de Recherche en Communications et en Cybernétique de Nantes (IRCCyN)
%+ Lab-STICC_ENSTAB_CID_IHSEV
%+ Pôle STIC_OSM
%+ Institut de recherche en astrophysique et planétologie (IRAP)
%+ ENAC Equipe MAIAA-OPTIM (MAIA-OPTIM)
%A Bourguignon, Sébastien
%A Ninin, Jordan
%A Carfantan, Hervé
%A Mongeau, Marcel
%Z GdR 720 ISIS
%< avec comité de lecture
%Z SIMS_HCERES2020
%@ 1053-587X
%J IEEE Transactions on Signal Processing
%I Institute of Electrical and Electronics Engineers
%V 64
%N 6
%P 1405-1419
%8 2016-03-01
%D 2016
%R 10.1109/TSP.2015.2496367
%K sparse approximation
%K optimization
%K L0-norm-based problems
%K mixed-integer programming
%K deconvolution
%Z Computer Science [cs]/Signal and Image ProcessingJournal articles
%X Sparse approximation addresses the problem of approximately fitting a linear model with a solution having as few non-zero components as possible. While most sparse estimation algorithms rely on suboptimal formulations, this work studies the performance of exact optimization of problems through Mixed-Integer Programs (MIPs). Nine different sparse optimization problems are formulated based on or data misfit measures, and involving whether constrained or penalized formulations. For each problem, MIP reformulations allow exact optimization, with optimality proof, for moderate-size yet difficult sparse estimation problems. Algorithmic efficiency of all formulations is evaluated on sparse deconvolution problems. This study promotes error-constrained minimization of the norm as the most efficient choice when associated with and misfits, while the misfit is more efficiently optimized with sparsity-constrained and sparsity-penalized problems. Then, exact optimization is shown to outperform classical methods in terms of solution quality, both for over-and under-determined problems. Finally, numerical simulations emphasize the relevance of the different fitting possibilities as a function of the noise statistical distribution. Such exact approaches are shown to be an efficient alternative, in moderate dimension, to classical (suboptimal) sparse approximation algorithms with data misfit. They also provide an algorithmic solution to less common sparse optimization problems based on and misfits. For each formulation, simulated test problems are proposed where optima have been successfully computed. Data and optimal solutions are made available as potential benchmarks for evaluating other sparse approximation methods.
%G English
%2 https://hal.archives-ouvertes.fr/hal-01254856/document
%2 https://hal.archives-ouvertes.fr/hal-01254856/file/MIP_sparse.pdf
%L hal-01254856
%U https://hal.archives-ouvertes.fr/hal-01254856
%~ ENAC
%~ MAIAA
%~ MAIAA-OPTIM
%~ CNRS
%~ UNIV-BREST
%~ UNIV-UBS
%~ EC-NANTES
%~ IRCCYN
%~ OMP-IRAP
%~ INSTITUT-TELECOM
%~ UNAM
%~ ENIB
%~ LAB-STICC_ENIB
%~ UNIV-NANTES
%~ LAB-STICC
%~ OMP
%~ ENSTA-BRETAGNE
%~ ENSTA-BRETAGNE-STIC
%~ IRCCYN-ADTSI
%~ OPTIM
%~ UNIV-TLSE3
%~ INSTITUTS-TELECOM
%~ INSU
%~ CNES
%~ METEO
%~ IRD
%~ ANR