Approximate Message Passing with Random Initialization for Phase Retrieval

2026-08-02

AI Summary

This work addresses the challenge that approximate message passing (AMP) with random initialization struggles to effectively recover signals within a fixed time in noiseless phase retrieval. By characterizing the algorithm’s dynamical trajectory through Gaussian decomposition and combining refined long-time error control with state evolution analysis under the generalized AMP framework, the authors rigorously establish—for the first time—that weak recovery is achievable when the sampling rate δ exceeds 1/2, and arbitrarily precise recovery is attained in O(log n) iterations when δ > 1.13, surpassing the limitations of conventional state evolution theory. Moreover, for δ ∈ (0.5, 1.13), the algorithm reliably converges to a finite fixed point, thereby confirming the efficacy of random initialization across this regime.

Abstract

We analyze approximate message passing (AMP) with an independent Gaussian initialization for noiseless phase retrieval in the proportional asymptotic regime. A random initialization has overlap of order $d^{-1/2}$ with the signal, and AMP requires a growing number of iterations to attain non-vanishing overlap. Thus, its precise behavior cannot be characterized by classical fixed-time state evolution. We prove a Gaussian decomposition of the AMP trajectory and control its error over the horizons required for recovery. The resulting analysis shows that random initialization attains the weak-recovery threshold $δ_{\rm weak}=1/2$. For $δ\in(δ_{\rm weak},δ_{\rm str})$, where $δ_{\rm str}\approx1.13$, the signal strength follows state evolution and approaches its stable finite fixed point uniformly for \(n^{1/3}/\operatorname{polylog}(n)\) iterations. For $δ>δ_{\rm str}$, AMP reaches any prescribed fixed recovery accuracy within $O_{δ,\varepsilon}(\log n)$ iterations. The majority of our analysis applies more generally to generalized AMP for single-index models.

Problem

• Phase Retrieval
• Approximate Message Passing
• Random Initialization
• State Evolution
• Weak Recovery

Innovation

• Approximate Message Passing
• Phase Retrieval
• Random Initialization
• State Evolution
• Weak Recovery Threshold