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In this paper, we aim to generalize the notion of restricted isometryconstant (RIC) in compressive sensing (CS) to restricted isometry randomvariable (RIV). Associated with a deterministic encoder there are two RICs,namely, the left and the right RIC. We show that these RICs can be generalizedto a left RIV and a right RIV for an ensemble of random encoders. We derive theprobability and the cumulative distribution functions of these RIVs for themost widely used i.i.d. Gaussian encoders. We also derive the asymptoticdistributions of the RIVs and show that the distribution of the left RIVconverges (in distribution) to the Weibull distribution, whereas that of theright RIV converges to the Gumbel distribution. By adopting the RIV framework,we bring to forefront that the current practice of using eigenvalues for RICprediction can be improved. We show on the one hand that the eigenvalue-basedapproaches tend to overestimate the RICs. On the other hand, the RIV-basedanalysis yields precise estimates of the RICs. We also demonstrate that thisprecise estimation aids to improve the previous RIC-based phase transitionanalysis in CS.

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