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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
75/2018

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

Gemma de las Cuevas, Tobias Fritz and Tim Netzer

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.

Received:
Aug 28, 2018
Published:
Aug 29, 2018
MSC Codes:
44A50, 47A10, 47A30, 12D15
Keywords:
moment problem, positive semidefinite, positive polynomial, sum of squares

Related publications

inJournal
2020 Repository Open Access
Gemma de las Cuevas, Tobias Fritz and Tim Netzer

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

In: Communications in mathematical physics, 375 (2020) 1, pp. 105-126