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There are well-known relationships between compressed sensing and thegeometry of the finite-dimensional $\ell_p$ spaces. A result of Kashin andTemlyakov can be described as a characterization of the stability of therecovery of sparse vectors via $\ell_1$-minimization in terms of the Gelfandwidths of certain identity mappings between finite-dimensional $\ell_1$ and$\ell_2$ spaces, whereas a more recent result of Foucart, Pajor, Rauhut andUllrich 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 theseresults: we characterize the stability of low-rank matrix recovery via Schatten$p$-(quasi-)norm minimization in terms of the Gelfand widths of certainidentity mappings between finite-dimensional Schatten $p$-spaces.

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