Compressive sensing bounds through a unifying framework for sparse models

In this work we investigate the sample complexity of support recovery in sparse signal processing models, with special focus on two compressive sensing scenarios. In particular, we consider models where N covariates X = (X₁,...,X_N) along with outcome Y are observed, with the assumption that the outcome Y is conditionally independent of the other covariates given K ≪ N covariates. Using asymptotic information theoretic analyses, we establish sufficient conditions on the number of samples in order to successfully recover the K salient covariates. We apply our results to two variants of the compressive sensing (CS) problem: (1) compressive sensing with a measurement noise model, (2) 1-bit quantized compressive sensing. In both models we consider sensing with independent and correlated Gaussian sensing matrices. We show that the sufficiency bounds we obtain on the number of measurements in both cases are comparable to the best known bounds while providing a novel perspective for the theoretical analysis of such models. In addition, we quantify how the correlation between the sensing columns affects the number of measurements. Our findings for the CS models demonstrate the applicability and flexibility of our general results on the sample complexity in sparse signal processing models.