On the PDF of the square of constrained minimal singular value for robust signal recovery analysis

In compressed sensing, the l₁-constrained minimal singular value (l₁-CMSV) of an encoder is used for analyzing (theoretically) the robustness of decoders against noise. In this paper, we show that for random encoders, the square of the l₁-CMSV (S-CMSV) is a random variable. And, for the Gaussian encoders, the S-CMSV admits a simple, closed-form probability and a cumulative distribution functions. We illustrate the benefits of these distributions for analyzing the robustness of various decoders. In particular, we interpret the existing theoretical robustness results of the decoders such as the basis pursuit in terms of the maximum possible undersampling.