An 𝓁0 Solution to Sparse Approximation Problems with Continuous Dictionaries
Megane Boudineau, Hervé Carfantan, Sébastien Bourguignon
Abstract
We address sparse approximation in the particular case where the dictionary is built upon the discretization of a continuous parameter. The resulting dictionary being highly correlated, equivalence between ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> and suboptimal solutions (e.g. greedy algorithms and convex relaxation) is not guaranteed. To tackle this issue, continuous parameter estimation has been proposed using a dictionary based on polar interpolation [1], [2]. Alternately, the exact ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> -norm optimization problem can be addressed on moderate size problems through Mixed Integer Programming (MIP) [3]. We propose to merge these two approaches in a new MIP formulation adapted to polar interpolation. Improvements on polar interpolation and refinements on its use in the ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -norm framework are also proposed. Methods are evaluated on simulated spike train deconvolution problems, where the proposed ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> -norm approach with continuous dictionary achieves the best results, although with higher computing time.
BibTeX
@inproceedings{icassp2018_an0solutiontospa,
title = {An 𝓁0 Solution to Sparse Approximation Problems with Continuous Dictionaries},
author = {Megane Boudineau and Hervé Carfantan and Sébastien Bourguignon},
booktitle = {ICASSP 2018},
year = {2018}
}