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

Abstract

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

$$n \times n$$

n × n

PSD matrices, denoted by D, using at most N number of

$$k \times k$$

k × 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

$$n \times n$$

n × n

symmetric matrices that are PSD when restricted to a fixed set of k-dimensional subspaces in

$${\mathbb {R}}^n$$

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

$$k = o(n)$$

k = o ( n )

. Second, we show that any set S that approximates D within a constant approximation ratio must have superpolynomial

$${\varvec{S}}_+^k$$

S

+ k

-extension complexity. To be more precise, if S is a constant factor approximation of D, then S must have

$${\varvec{S}}_+^k$$

S

+ k

-extension complexity at least

$$\exp ( C \cdot \min \{ \sqrt{n}, n/k \})$$

exp ( C · min

{

n

, n / k }

)

where C is some absolute constant. In addition, we show that any set S such that

$$D \subseteq S$$

D ⊆ S

and the Gaussian width of S is at most a constant times larger than the Gaussian width of D must have

$${\varvec{S}}_+^k$$

S

+ k

-extension complexity at least

$$\exp ( C \cdot \min \{ n^{1/3}, \sqrt{n/k} \})$$

exp ( C · min

{

n

1 / 3

,

n / k

}

)

. These results imply that the cone of

$$n \times n$$

n × n

PSD matrices cannot be approximated by a polynomial number of

$$k \times k$$

k × k

PSD constraints for any

$$k = o(n / \log ^2 n)$$

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

$${\varvec{S}}_+^n$$

S

+ n

, which corresponds to the special case with

$$k=1$$

k = 1

.