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arXiv · 2303.00844

The greedy side of the LASSO: New algorithms for weighted sparse recovery via loss function-based orthogonal matching pursuit

Sina Mohammad-Taheri·Simone Brugiapaglia
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Abstract

We propose a class of greedy algorithms for weighted sparse recovery by considering new loss function-based generalizations of Orthogonal Matching Pursuit (OMP). Given a (regularized) loss function, the proposed algorithms alternate the iterative construction of the signal support via greedy index selection and a signal update based on solving a local data-fitting problem restricted to the current support. We show that greedy selection rules associated with popular weighted sparsity-promoting loss functions admit explicitly computable and simple formulas. Specifically, we consider $ \ell^0 $- and $ \ell^1 $-based versions of the weighted LASSO (Least Absolute Shrinkage and Selection Operator), the Square-Root LASSO (SR-LASSO) and the Least Absolute Deviations LASSO (LAD-LASSO). Through numerical experiments on Gaussian compressive sensing and high-dimensional function approximation, we demonstrate the effectiveness of the proposed algorithms and empirically show that they inherit desirable characteristics from the corresponding loss functions, such as SR-LASSO's noise-blind optimal parameter tuning and LAD-LASSO's fault tolerance. In doing so, our study sheds new light on the connection between greedy sparse recovery and convex relaxation.

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Sina Mohammad-Taheri, Simone Brugiapaglia. 2023-03-01. The greedy side of the LASSO: New algorithms for weighted sparse recovery via loss function-based orthogonal matching pursuit. https://doi.org/10.1007/s43670-025-00098-0

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