Effect of global shrinkage parameter of horseshoe prior in compressed sensing

In sparse signal processing, this study investigates the effect of the global shrinkage parameter $\tau$ of a horseshoe prior, one of the global-local shrinkage prior, on the linear regression. Statistical mechanics methods are employed to examine the accuracy of signal estimation. A phase diagram of successful and failure of signal recovery in noise-less compressed sensing with varying $\tau$ is discussed from the viewpoint of dynamic characterization of the approximate message passing as a solving algorithm and static characterization of the free-energy landscape. It is found that there exists a parameter region where the approximate message passing algorithm can hardly recover the true signal, even though the true signal is locally stable. The analysis of the free-energy landscape also provides important insight into the optimal choice of $\tau$.