Random Tight Frames

TL;DR

This paper introduces probabilistic frames to analyze random finite frames, showing that various random matrices used in compressed sensing can be viewed as probabilistic tight frames, thus broadening the understanding of frame theory in random settings.

Contribution

It develops probabilistic versions of tight frames and FUNTFs, relaxing previous strict conditions and linking them to random matrices in compressed sensing.

Findings

Random points on the sphere form approximate finite unit norm tight frames.

Probabilistic tight frames can be chosen without identical distribution or unit norm constraints.

Classes of random matrices in compressed sensing are induced by probabilistic tight frames.

Abstract

We introduce probabilistic frames to study finite frames whose elements are chosen at random. While finite tight frames generalize orthonormal bases by allowing redundancy, independent, uniformly distributed points on the sphere approximately form a finite unit norm tight frame (FUNTF). In the present paper, we develop probabilistic versions of tight frames and FUNTFs to significantly weaken the requirements on the random choice of points to obtain an approximate finite tight frame. Namely, points can be chosen from any probabilistic tight frame, they do not have to be identically distributed, nor have unit norm. We also observe that classes of random matrices used in compressed sensing are induced by probabilistic tight frames.