Absence of a hard phase in BSR with Gaussian label noise

Prove that no computational-to-statistical gap (hard phase) occurs in Bayes-optimal inference for the bilinear sequence regression (BSR) model with factorised Gaussian prior and Gaussian label noise; equivalently, show that the GAMP-RIE algorithm converges from zero-overlap initialization to the Bayes-optimal fixed point across the full parameter regime without encountering suboptimal fixed points.

Background

The paper develops a GAMP-RIE algorithm and state evolution for the BSR model and shows that for the specific case of factorised Gaussian priors with Gaussian label noise, numerical experiments and free-entropy analysis do not indicate a computational-to-statistical gap. In such gaps, AMP-type algorithms fail to reach the Bayes-optimal estimator despite information-theoretic recoverability.

Establishing the absence of a hard phase would provide a rigorous guarantee that GAMP-RIE achieves the Bayes-optimal mean-squared error for the BSR model with Gaussian label noise, resolving a central algorithmic optimality question raised by the authors’ empirical findings.