Geometric Perspective on Concentration Phenomena in Frame Theory

Abstract: Frames are fundamental structures in many areas, and tight frames are particularly valued for their stability and robustness properties. In this work, we establish concentration phenomena for Parseval frames, i.e. tight frames with frame bound 1, under isotropic distributions supported on the sphere and the Euclidean ball, showing that epsilon-nearly Parseval frames are prevalent in these probabilistic models. We further introduce a distinguished subclass of Parseval frames and prove that they are both robust under the Bernoulli-type erasure model and prevalent within the space of Parseval frames. As an application of our results, we derive a high-probability upper bound of order o(epsilon d) for the Paulsen problem.