Factorization of positive-semidefinite operators with absolutely summable entries

Abstract

A problem by Feichtinger, Heil, and Larson asks whether every infinite matrix $A$ with ${\sum }_{k,l}|A_{kl}|<\infty$ (an equivalent substitute for the Feichtinger algebra) that is positive-semidefinite admits a symmetric rank-one decomposition $A={\sum }_k{f}_k^{\ast }\otimes f_k$ with ${\sum }_k\Vert f_k{\Vert }_1^2<\infty$. In the finite-dimensional setting, we analyze the corresponding quantitative ${\ell }_1^n$ optimization problem by an exact reformulation as a linear program over measures, derive its dual, and prove strong duality. We then obtain an equivalent adjoint formulation regarding the quality of a convex relaxation. In the infinite-dimensional setting, we first provide a negative answer to this question using a concurrent finite-dimensional result by Bandeira-Mixon-Steinerberger. We further study the collection of operators for which such decomposition exists, showing that they are dense in a suitable topology and invariant under the action of the positive-coefficient analytic Wiener subalgebra. In addition, we give a sufficient condition for successful rank-one decomposition in terms of $2$-summing factorization, and we characterize exactly when $A^{1/2}$ is $2$-summing.

27 pages, significant update in the infinite-dimensional settings. Abstract update