2014/06/30 by Javier Alejandro Chávez‐Domínguez, Javier Alejandro Chávez-Domínguez, Denka Kutzarova
Computer Science · Engineering · Mathematics · #Banach space #Characterization (materials science) #Combinatorics #Computer science #Eigenvalues and eigenvectors #Mathematical Analysis and Transform Methods #Mathematics #Matrix (chemical analysis) #Matrix norm #Microwave Imaging and Scattering Analysis #Noncommutative geometry #Norm (philosophy) #Pure mathematics #Rank (graph theory) #Space (punctuation) #Sparse and Compressive Sensing Techniques #Stability (learning theory) #cs.IT #math.FA #math.IT
paper · pdf · doi:10.1016/j.jmaa.2015.02.041
published as J. Math. Anal. Appl. 427 (2015), no. 1, 320--335 · 16 pages
openalex publication_date 2015/02/17 · arxiv created 2015/06/19 · arxiv updated 2015/06/22 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
There are well-known relationships between compressed sensing and the geometry of the finite-dimensional ℓp spaces. A result of Kashin and Temlyakov can be described as a characterization of the stability of the recovery of sparse vectors via ℓ1-minimization in terms of the Gelfand widths of certain identity mappings between finite-dimensional ℓ1 and ℓ2 spaces, whereas a more recent result of Foucart, Pajor, Rauhut and Ullrich proves an analogous relationship even for ℓp spaces with p < 1. In this paper we prove what we call matrix or noncommutative versions of these results: we characterize the stability of low-rank matrix recovery via Schatten p-(quasi-)norm minimization in terms of the Gelfand widths of certain identity mappings between finite-dimensional Schatten p-spaces.