Support Recovery with Sparsely Sampled Free Random Matrices

Consider a Bernoulli-Gaussian complex $n$-vector whose components are $V_i = X_i B_i$, with X_i \sim \Cc\Nc(0,\Pc_x) and binary $B_i$ mutually independent and iid across $i$. This random $q$-sparse vector is multiplied by a square random matrix \Um, and a randomly chosen subset, of average size $n p$, $p \in [0,1]$, of the resulting vector components is then observed in additive Gaussian noise. We extend the scope of conventional noisy compressive sampling models where \Um is typically %A16 the identity or a matrix with iid components, to allow \Um satisfying a certain freeness condition. This class of matrices encompasses Haar matrices and other unitarily invariant matrices. We use the replica method and the decoupling principle of Guo and Verd\'u, as well as a number of information theoretic bounds, to study the input-output mutual information and the support recovery error rate in the limit of $n \to \infty$. We also extend the scope of the large deviation approach of Rangan, Fletcher and Goyal and characterize the performance of a class of estimators encompassing thresholded linear MMSE and $\ell_1$ relaxation

RSB Decoupling Property of MAP Estimators

The large-system decoupling property of a MAP estimator is studied when it estimates the i.i.d. vector $\boldsymbol{x}$ from the observation $\boldsymbol{y}=\mathbf{A}\boldsymbol{x}+\boldsymbol{z}$ with $\mathbf{A}$ being chosen from a wide range of matrix ensembles, and the noise vector $\boldsymbol{z}$ being i.i.d. and Gaussian. Using the replica method, we show that the marginal joint distribution of any two corresponding input and output symbols converges to a deterministic distribution which describes the input-output distribution of a single user system followed by a MAP estimator. Under the $b$RSB assumption, the single user system is a scalar channel with additive noise where the noise term is given by the sum of an independent Gaussian random variable and $b$ correlated interference terms. As the $b$RSB assumption reduces to RS, the interference terms vanish which results in the formerly studied RS decoupling principle.Comment: 5 pages, presented in Information Theory Workshop 201