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Analysis of the ratio of ℓ1 and ℓ2 norms in compressed sensing
Applied and Computational Harmonic Analysis  (IF3.055),  Pub Date : 2021-07-08, DOI: 10.1016/j.acha.2021.06.006
Yiming Xu, Akil Narayan, Hoang Tran, Clayton G. Webster

We study the ratio of 1 and 2 norms (1/2) as a sparsity-promoting objective in compressed sensing. We first propose a novel criterion that guarantees that an s-sparse signal is the local minimizer of the 1/2 objective; our criterion is interpretable and useful in practice. We also give the first uniform recovery condition using a geometric characterization of the null space of the measurement matrix, and show that this condition is satisfied for a class of random matrices. We also present analysis on the robustness of the procedure when noise pollutes data. Numerical experiments are provided that compare 1/2 with some other popular non-convex methods in compressed sensing. Finally, we propose a novel initialization approach to accelerate the numerical optimization procedure. We call this initialization approach support selection, and we demonstrate that it empirically improves the performance of existing 1/2 algorithms.