Geometric Perspective on Concentration Phenomena in Frame Theory

TL;DR

This paper provides geometric concentration bounds demonstrating that random frames are nearly Parseval or equal-norm with high probability, offering new probabilistic bounds for the Paulsen problem.

Contribution

It introduces non-asymptotic concentration bounds for random frames using geometric measure concentration on Riemannian manifolds, with applications to the Paulsen problem.

Findings

Random equal-norm frames are nearly Parseval with high probability.

Random Parseval frames are nearly equal-norm with high probability.

A new probabilistic upper bound for the Paulsen problem is established.

Abstract

Parseval and equal-norm frames play a fundamental role in frame theory and signal processing. In this work, we prove non-asymptotic concentration bounds showing that random equal-norm frames are nearly Parseval with high probability, and that random Parseval frames are nearly equal-norm with high probability. Our proofs are geometric in nature, and rely on general measure concentration principles in Riemannian manifolds. As an application, we obtain a novel probabilistic upper bound for the Paulsen problem.