On Approximations of the PSD Cone by a Polynomial Number of Smaller-sized PSD Cones

TL;DR

This paper establishes fundamental lower bounds on approximating the cone of positive semidefinite matrices using smaller PSD cones, showing such approximations require exponentially many constraints unless the size of the smaller cones is proportional to the original matrix size.

Contribution

It proves that approximating the PSD cone with polynomially many small PSD cones is impossible unless the small cones are nearly as large as the original matrices, extending prior hardness results.

Findings

Any good approximation requires exponentially many subspaces.

Superpolynomial extension complexity is necessary for constant-factor approximations.

Approximation by polynomially many small PSD cones is impossible for small k.

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 $\mathbb{RR}^n$. 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 $\mathbf{S}_+^k$-extension complexity. To be more precise, if $S$ is a constant factor approximation of $D$, then $S$ must have $\mathbf{S}_+^k$-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 $D$ is at most a constant times larger than the Gaussian width of $D$ must have $\mathbf{S}_+^k$-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 on the hardness of polyhedral approximations of $\mathbf{S}_+^n$, which corresponds to the special case with $k=1$.