Compressed sensing with an a priori distribution

Abstract

Compressed sensing is a technique for recovering an unknown sparse signal from a number of random linear measurements. The number of measurements required for perfect recovery plays a key role and it exhibits a phase transition. If the number of measurements exceeds certain level related with the sparsity of the signal, exact recovery is obtained with high probability. If the number of measurements is below this level, exact recovery occurs with very small probability. In this work we are able to reduce this threshold by incorporating statistical information about the data we wish to recover. Our algorithm works by minimizing a suitably weighted 11-norm, where the weights are chosen so that the expected statistical dimension of the descent cones of a weighted cross-polytope is minimized. We also provide Monte Carlo algorithms for computing intrinsic volumes of these descent cones and estimating the failure probability of our methods