AI 中文总结
该研究分析了带随机高斯初始化的近似消息传递（AMP）在无噪相位恢复中的性能，证明其轨迹的高斯分解并控制误差，明确了不同参数区间的恢复阈值与迭代复杂度，结果可推广至单指标模型的广义AMP。
AI 中文摘要
我们在比例渐近情形下分析了带独立高斯初始化的近似消息传递（AMP）用于无噪相位恢复的性能。随机初始化与信号的重叠阶为$d^{-1/2}$，AMP需要不断增加的迭代次数才能获得非零重叠，因此其精确行为无法用经典的固定时间状态演化来刻画。我们证明了AMP轨迹的高斯分解，并在恢复所需的时间范围内控制其误差。分析结果表明，随机初始化可达到弱恢复阈值$δ_{\rm weak}=1/2$。当$δ∈(δ_{\rm weak},δ_{\rm str})$（其中$δ_{\rm str}≈1.13$）时，信号强度遵循状态演化，且在$n^{1/3}/\text{polylog}(n)$次迭代内一致趋近于其稳定有限不动点。当$δ>δ_{\rm str}$时，AMP可在$O_{δ,ε}(\text{log}n)$次迭代内达到任意指定的固定恢复精度。我们的大部分分析可更广泛地应用于单指标模型的广义AMP。
英文摘要
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.
