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Méthodes de programmation en nombres mixtes pour l'optimisation parcimonieuse en traitement du signal

Abstract : Sparse approximation aims to fit a linear model in a least-squares sense, with a small number of non-zero components (the L0 “norm”). Due to its combinatorial nature, it is often addressed by suboptimal methods. It was recently shown, however, that exact resolution could be performed through a mixed integer program(MIP) reformulation solved by a generic solver, implementing branch-and-bound techniques. This thesis addresses the L0-norm sparse approximation problem with tailored branch-andbound resolution methods, exploiting the mathematical structures of the problem. First, we show that each node evaluation amounts to solving an L1-norm problem, for which we propose dedicated methods. Then, we build an efficient exploration strategy exploiting the sparsity of the solution, by activating first the non-zero variables in the tree search. The proposed method outperforms the CPLEX solver, reducing the computation time and making it possible to address larger problems. In a second part of the thesis, we propose and study the MIP reformulations of the spectral unmixing problem with L0-norm sparsity more advanced structured sparsity constraints, which are usually addressed through relaxations in the literature. We show that, for problems with limited complexity (highly sparse solutions, good signal-to-noise ratio), such constraints can be accounted for exactly and improve the estimation quality over standard approaches.
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Submitted on : Wednesday, May 26, 2021 - 4:55:11 PM
Last modification on : Wednesday, April 27, 2022 - 4:41:58 AM
Long-term archiving on: : Friday, August 27, 2021 - 8:36:25 PM


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  • HAL Id : tel-03237601, version 1


Ramzi Ben Mhenni. Méthodes de programmation en nombres mixtes pour l'optimisation parcimonieuse en traitement du signal. Traitement du signal et de l'image [eess.SP]. École centrale de Nantes, 2020. Français. ⟨NNT : 2020ECDN0008⟩. ⟨tel-03237601⟩



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