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 p spaces. A result of Kashin and Temlyakov [20] can be described as a characterization of the stability of the recovery of sparse vectors via 1 -minimization in terms of the Gelfand widths of certain identity mappings between finite-dimensional 1 and 2 spaces, whereas a more recent result of Foucart, Pajor, Rauhut and Ullrich [16] proves an analogous relationship even for 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.

Entropy numbers of embeddings of Schatten classes

“…Here p,q refers to equivalence up to constants depending only on p and q. As we pointed out before, their result has very nice applications in the theory of low-rank matrix recovery and we refer the reader to [11] and the references cited therein for a detailed discussion. The result of Chávez-Domínguez and Kutzarova, and in particular the one of Carl and Defant, is complemented by asymptotic bounds obtained by Hinrichs and Michels [26].…”

“…Recent years have seen increased interest in those non-commutative L p spaces. One such case, in part motivating our work, lies in the area of compressed sensing in the form of low-rank matrix recovery (see, e.g., [4,5,8] and the references therein. )…”

“…As typical in the theory of Banach spaces, our proofs reflect a variety of ideas and tools of analytical and geometric flavor. These include • Carl's inequality for the case of quasi-Banach spaces [13] • two-sided asymptotic estimates for the Gelfand numbers of identities of Schatten classes [8] • volume estimates for metric balls in the Grassmannian going back to the work of Szarek [24] (see also [17]), and…”

“…Recent years have again seen an increased interest in Schatten trace classes in parts because of their role in low-rank matrix recovery, the non-commutative analogue to the classical compressed sensing approach. We refer the reader to [11,17] and the references cited therein for more information.…”

“…where we implicitly used the equality of approximation and Gelfand numbers for operators defined on a Hilbert space [42] (see also [22,Lemma 1.2] for their equivalence in the case of type 2 spaces). Two decades later, following ideas of Foucart, Pajor, Rauhut, and Ullrich from [16], Chávez-Domínguez and Kutzarova obtained an asymptotic formula in the quasi-Banach space setting [11], proving that…”