Restricted Isometry Random Variables: Probability Distributions, RIC Prediction and Phase Transition Analysis for Gaussian Encoders

Abstract

In this paper, we aim to generalize the notion of restricted isometry constant (RIC) in compressive sensing (CS) to restricted isometry random variable (RIV). Associated with a deterministic encoder there are two RICs, namely, the left and the right RIC. We show that these RICs can be generalized to a left RIV and a right RIV for an ensemble of random encoders. We derive the probability and the cumulative distribution functions of these RIVs for the most widely used i.i.d. Gaussian encoders. We also derive the asymptotic distributions of the RIVs and show that the distribution of the left RIV converges (in distribution) to the Weibull distribution, whereas that of the right RIV converges to the Gumbel distribution. By adopting the RIV framework, we bring to forefront that the current practice of using eigenvalues for RIC prediction can be improved. We show on the one hand that the eigenvalue-based approaches tend to overestimate the RICs. On the other hand, the RIV-based analysis yields precise estimates of the RICs. We also demonstrate that this precise estimation aids to improve the previous RIC-based phase transition analysis in CS.