Geometric Perspective on Concentration Phenomena in Frame Theory

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.