On approximations of the PSD cone by a polynomial number of smaller-sized PSD cones

Abstract

We study the problem of approximating the cone of positive semidefinite (PSD) matrices with a cone that can be described by smaller-sized PSD constraints. Specifically, we ask the question: “how closely can we approximate the set of unit-trace \(n \times n\) PSD matrices, denoted by D, using at most N number of \(k \times k\) PSD constraints?” In this paper, we prove lower bounds on N to achieve a good approximation of D by considering two constructions of an approximating set. First, we consider the unit-trace \(n \times n\) symmetric matrices that are PSD when restricted to a fixed set of k-dimensional subspaces in \({\mathbb {R}}^n\). We prove that if this set is a good approximation of D, then the number of subspaces must be at least exponentially large in n for any \(k = o(n)\). Second, we show that any set S that approximates D within a constant approximation ratio must have superpolynomial \({\varvec{S}}_+^k\)-extension complexity. To be more precise, if S is a constant factor approximation of D, then S must have \({\varvec{S}}_+^k\)-extension complexity at least \(\exp ( C \cdot \min \{ \sqrt{n}, n/k \})\) where C is some absolute constant. In addition, we show that any set S such that \(D \subseteq S\) and the Gaussian width of S is at most a constant times larger than the Gaussian width of D must have \({\varvec{S}}_+^k\)-extension complexity at least \(\exp ( C \cdot \min \{ n^{1/3}, \sqrt{n/k} \})\). These results imply that the cone of \(n \times n\) PSD matrices cannot be approximated by a polynomial number of \(k \times k\) PSD constraints for any \(k = o(n / \log ^2 n)\). These results generalize the recent work of Fawzi (Math Oper Res 46(4):1479–1489, 2021) on the hardness of polyhedral approximations of \({\varvec{S}}_+^n\), which corresponds to the special case with \(k=1\).