Selector form of Weaver's conjecture, Feichtinger's conjecture, and frame sparsification

TL;DR

This paper extends probabilistic and selector results related to Weaver's and Feichtinger's conjectures, providing new bounds, solutions to open problems, and generalizations for frames and Bessel sequences in infinite-dimensional settings.

Contribution

It introduces novel selector theorems for block diagonal positive semidefinite matrices, extending key conjectures and solving open problems in frame theory and operator measures.

Findings

Extended probabilistic results for block diagonal matrices.

Provided selector forms for Weaver's and Feichtinger's conjectures.

Achieved nearly tight discretization of continuous frames.

Abstract

We show an extension of a probabilistic result of Marcus, Spielman, and Srivastava, which resolved the Kadison-Singer problem, for block diagonal positive semidefinite random matrices. We use this result to show several selector results, which generalize their partition counterparts. This includes a selector form of Weaver's KS$_r$ conjecture for block diagonal trace class operators, which extends a selector result for Bessel sequences, or equivalently rank one matrices, due to Londner and the author. We also show a selector variant of Feichtinger's conjecture for a (possibly infinite) collection of Bessel sequences, extending earlier results for a single Bessel sequence. We prove a generalization of the $R_\epsilon$ conjecture of Casazza, Tremain, and Vershynin for infinite collection of equal norm Bessel sequences. In particular, our selector result yields a conjectured asymptotically optimal bound for a single Bessel sequence in terms of Riesz sequence tightness parameter. We establish an iterated selector form of Weaver's KS$_2$ conjecture and show its applications. This includes a solution of an open problem on nearly unit norm Parseval frames of exponentials, which was posed by Londner and the author. We generalize a discretization result for continuous frames by Freeman and Speegle in two ways. First, we extend their result from the setting of rank one operators to positive trace operator valued measures. Second, we establish a nearly tight discretization of bounded continuous Parseval frames. In particular, our selector result yields an improvement of the result of Nitzan, Olevskii, and Ulanovskii and implies the existence of nearly tight exponential frames for unbounded sets with an explicit control on their frame redundancy.