Universal MAP estimation in compressed sensing

We study the compressed sensing (CS) estimation problem where an input is measured via a linear matrix multiplication under additive noise. While this setup usually assumes sparsity or compressibility in the observed signal during recovery, the signal structure that can be leveraged is often not known a priori. In this paper, we consider universal CS recovery, where the statistics of a stationary ergodic signal source are estimated simultaneously with the signal itself. We provide initial theoretical, algorithmic, and experimental evidence based on maximum a posteriori (MAP) estimation that shows the promise of universality in CS, particularly for low-complexity sources that do not exhibit standard sparsity or compressibility.