Stability of low-rank matrix recovery and its connections to Banach space geometry

Abstract: There are well-known relationships between compressed sensing and the geometry of the finite-dimensional $\ell_p$ spaces. A result of Kashin and Temlyakov can be described as a characterization of the stability of the recovery of sparse vectors via $\ell_1$-minimization in terms of the Gelfand widths of certain identity mappings between finite-dimensional $\ell_1$ and $\ell_2$ spaces, whereas a more recent result of Foucart, Pajor, Rauhut and Ullrich proves an analogous relationship even for $\ell_p$ 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.