Phase transition in compressed sensing with horseshoe prior

In Bayesian statistics, horseshoe prior has attracted increasing attention as an approach to the sparse estimation. The estimation accuracy of compressed sensing with the horseshoe prior is evaluated by statistical mechanical method. It is found that there exists a phase transition in signal recoverability in the plane of the number of observations and the number of nonzero signals and that the recoverability phase is more extended than that using the well-known $l_1$ norm regularization.