Effect of global shrinkage parameter of horseshoe prior in compressed sensing

This study investigates the effect of the global shrinkage parameter τ of a horseshoe prior, one of the global–local shrinkage priors, on linear regression in sparse signal processing. Statistical mechanics methods are employed to examine the accuracy of signal estimation. The phase diagram of the success and failure of signal recovery in noiseless compressed sensing with varying τ is discussed from the viewpoint of dynamic characterization of approximate message passing (AMP) as a solving algorithm and static characterization of the free-energy landscape. It is found that there exists a parameter region where the AMP 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 τ.