Effect of global shrinkage parameter of horseshoe prior in compressed sensing

Abstract

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 $\tau$.

arXiv comment13 pages, 15 figures

References in corpus (9)

• Observed Universality of Phase Transitions in High-Dimensional Geometry, with Implications for Modern Data Analysis and Signal Processing
• Probabilistic Reconstruction in Compressed Sensing: Algorithms, Phase Diagrams, and Threshold Achieving Matrices
• Statistical physics-based reconstruction in compressed sensing
• A typical reconstruction limit of compressed sensing based on Lp-norm minimization
• The Horseshoe Estimator: Posterior Concentration around Nearly Black Vectors
• Asymptotic Bayes-optimality under sparsity of some multiple testing procedures
• Conditions for Posterior Contraction in the Sparse Normal Means Problem
• Statistical mechanical analysis of sparse linear regression as a variable selection problem
• Phase transition in compressed sensing with horseshoe prior