Optimal bounds on the positivity of a matrix from a few moments

TL;DR

This paper develops asymptotically optimal bounds for assessing the positivity of large Hermitian matrices using only a few moments, with practical methods and applications in tensor networks.

Contribution

It introduces new bounds on matrix positivity from limited moments and compares three computational methods for these bounds, including SDP and linear relaxations.

Findings

The bounds are asymptotically optimal in Schatten p-norms.

Three computational methods are analyzed: sos polynomial, Handelman, and Chebyshev.

Applications include tensor networks and free spectrahedra.

Abstract

In many contexts one encounters Hermitian operators $M$ on a Hilbert space whose dimension is so large that it is impossible to write down all matrix entries in an orthonormal basis. How does one determine whether such $M$ is positive semidefinite? Here we approach this problem by deriving asymptotically optimal bounds to the distance to the positive semidefinite cone in Schatten $p$-norm for all integer $p\in[1,\infty)$, assuming that we know the moments $\mathbf{tr}(M^k)$ up to a certain order $k=1,\ldots, m$. We then provide three methods to compute these bounds and relaxations thereof: the sos polynomial method (a semidefinite program), the Handelman method (a linear program relaxation), and the Chebyshev method (a relaxation not involving any optimization). We investigate the analytical and numerical performance of these methods and present a number of example computations, partly motivated by applications to tensor networks and to the theory of free spectrahedra.