Sparse Recovery from Group Orbits

Abstract: While most existing sparse recovery results allow only minimal structure within the measurement scheme, many practical problems possess significant structure. To address this gap, we present a framework for structured measurements that are generated by random orbits of a group representation associated with a finite group. We differentiate between two scenarios: one in which the sampling set is fixed and another in which the sampling set is randomized. For each case, we derive an estimate for the number of measurements required to ensure that the restricted isometry property holds with high probability. These estimates are contingent upon the specific representation employed. For this reason, we analyze and characterize various representations that yield favorable recovery outcomes, including the left regular representation. Our work not only establishes a comprehensive framework for sparse recovery of group-structured measurements but also generalizes established measurement schemes, such as those derived from partial random circulant matrices.