Convergence and optimality of the linearized AMP fixed point

Determine whether the linearized approximate message passing (AMP) scheme with normalized linear denoiser g_t(x) = x / sqrt(1 + μ_t^2), applied to the Fisher matrix S_{k_F}, provably converges to the non-zero fixed point in the weak-recovery regime (1/Δ_{k_F} > 1/m_{2k_F}^2), and ascertain whether that non-trivial fixed point coincides with the maximizer of the replica-symmetric functional F(Δ_{k_F}, q) that characterizes the Bayes-optimal limit.

Background

The authors derive a linearized AMP method that approximates a spectral algorithm via a specific choice of linear denoiser and show that its phase transition matches the information-theoretic recovery threshold. However, they note the absence of guarantees about convergence to the non-zero fixed point and whether that fixed point equals the replica-symmetric maximizer.

Resolving these questions would clarify the algorithmic optimality of linearized AMP relative to the Bayes-optimal characterization and strengthen the theoretical understanding of spectral and AMP-based recovery in the higher-order Fisher universality setting.