模型失配场景下向量近似消息传递的宏观分析
Macroscopic Analysis of Vector Approximate Message Passing in a Model Mismatch Setting
AI总结：
本文将VAMP的性能分析拓展至模型失配场景，推导了其状态演化方程，证明不动点与复制对称解一致且存在微观不稳定性，数值实验验证了结论。
摘要：
向量近似消息传递（Vector approximate message passing, VAMP）是一种用于广义线性模型的高效近似推理算法。尽管VAMP表现出优异性能，尤其是当测量矩阵采样自旋转不变系综时，但现有收敛性与性能分析大多局限于可获取正确后验分布的情形。本文将分析拓展至推理阶段未使用正确后验分布的场景。我们推导了宏观描述VAMP动力学的状态演化方程，并证明其不动点与统计力学中复制方法得到的复制对称解一致。我们还表明VAMP的不动点会出现微观不稳定性，其临界条件与复制对称破缺的临界条件相符。数值实验结果支持了我们的结论。
英文摘要：
Vector approximate message passing (VAMP) is an efficient approximate inference algorithm used for generalized linear models. Although VAMP exhibits excellent performance, particularly when measurement matrices are sampled from rotationally invariant ensembles, existing convergence and performance analyses have been limited mostly to cases in which the correct posterior distribution is available. Here, we extend the analyses for cases in which the correct posterior distribution is not used in the inference stage. We derive state evolution equations, which macroscopically describe the dynamics of VAMP, and show that their fixed point is consistent with the replica symmetric solution obtained by the replica method of statistical mechanics. We also show that the fixed point of VAMP can exhibit a microscopic instability, the critical condition of which agrees with that for breaking the replica symmetry. The results of numerical experiments support our findings.
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