Analysis of the ratio of ℓ1 and ℓ2 norms in compressed sensing

Yiming Xu, Akil Narayan, Hoang Tran, Clayton G. Webster

Research output: Contribution to journalArticlepeer-review

15 Scopus citations

Abstract

We study the ratio of ℓ1 and ℓ2 norms (ℓ1/ℓ2) 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 ℓ1/ℓ2 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 ℓ1/ℓ2 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 ℓ1/ℓ2 algorithms.

Original languageEnglish
Pages (from-to)486-511
Number of pages26
JournalApplied and Computational Harmonic Analysis
Volume55
DOIs
StatePublished - Nov 2021

Funding

We would like to thank the anonymous referees for their very helpful comments which significantly improve the presentation of the paper. The first author ( [email protected] ) thanks Tom Alberts, You-Cheng Chou and Dong Wang for constructive discussions. The first and second authors ( [email protected] , [email protected] ) acknowledge partial support from NSF DMS-1848508 . The third author ( [email protected] ) acknowledges support from Scientific Discovery through Advanced Computing (SciDAC) program through the FASTMath Institute under Contract No. DE-AC02-05CH11231 . The last author ( [email protected] ) acknowledges the U.S. Department of Energy , Office of Science, Early Career Research Program under award number ERKJ314 ; U.S. Department of Energy , Office of Advanced Scientific Computing Research under award numbers ERKJ331 and ERKJ345 ; and the National Science Foundation , Division of Mathematical Sciences, Computational Mathematics program under contract number DMS1620280 .

Keywords

  • Compressed sensing
  • High-dimensional geometry
  • Non-convex optimization
  • Random matrices

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