On approximations of the PSD cone by a polynomial number of smaller-sized PSD cones
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Author(s)
Song, Dogyoon
Parrilo, Pablo A.
Date Issued
April 1, 2022
Publisher
Springer Berlin Heidelberg
Citation
Song, Dogyoon and Parrilo, Pablo A. 2022. "On approximations of the PSD cone by a polynomial number of smaller-sized PSD cones."
Version
Author's final manuscript
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
.
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
Massachusetts Institute of Technology. Laboratory for Information and Decision Systems
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DOI of Published Version
https://doi.org/10.1007/s10107-022-01795-7
