The Restricted Isometry Property for Measurements from Group Orbits

TL;DR

This paper extends the analysis of sparse recovery guarantees from random circulant matrices to measurements derived from group orbits, providing new bounds on measurement requirements based on group representation theory.

Contribution

It introduces a generalized framework for analyzing the restricted isometry property for measurements from group orbits, incorporating representation-theoretic assumptions.

Findings

Derived measurement bounds for RIP using group orbit measurements

Highlighted importance of representation-theoretic conditions

Provided examples illustrating theoretical results

Abstract

It is known that sparse recovery by measurements from random circulant matrices provides good recovery bounds. We generalize this to measurements that arise as a random orbit of a group representation for some finite group G. We derive estimates for the number of measurements required to guarantee the restricted isometry property with high probability. Following this, we present several examples highlighting the role of appropriate representation-theoretic assumptions.