APPROXIMATE MESSAGE PASSING ALGORITHMS FOR ROTATIONALLY INVARIANT MATRICES
APPROXIMATE MESSAGE PASSING ALGORITHMS FOR ROTATIONALLY INVARIANT MATRICES
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DOI：
10.1214/21-aos2101
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发表时间：
2022-02-01
影响因子：
4.5
通讯作者：
Fan, Zhou
中科院分区：
文献类型：
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作者：
Fan, Zhou
近似消息传递（AMP）算法已经在各种应用中得到广泛使用。然而，其Onsager校正和状态演化的精确形式取决于底层随机矩阵系综的性质，限制了针对白色噪声导出的AMP算法可适用于实际中出现的数据矩阵的程度。在这项工作中，我们研究了随机矩阵W满足正交旋转不变性定律的更一般的AMP算法，其中W可能具有与白色噪声特征的半圆和马森科-帕斯托尔定律不同的光谱分布。这些算法中的Onsager修正和状态演化由W的谱分布的自由累积量或矩形自由累积量定义。他们的形式来自以前的Opper，Cakmak和Winther使用非严格的动态泛函理论技术，我们提供了严格的proofs.Our激励应用程序是主成分分析的贝叶斯-AMP算法，当有先验结构的主成分（PC）和可能的非白噪声。对于足够大的信号强度和任何非高斯先验分布的PC，我们证明了该算法可证明达到更高的估计精度比样本PC。
Approximate Message Passing (AMP) algorithms have seen widespread use across a variety of applications. However, the precise forms for their Onsager corrections and state evolutions depend on properties of the underlying random matrix ensemble, limiting the extent to which AMP algorithms derived for white noise may be applicable to data matrices that arise in practice.In this work, we study more general AMP algorithms for random matrices W that satisfy orthogonal rotational invariance in law, where W may have a spectral distribution that is different from the semicircle and Marcenko- Pastur laws characteristic of white noise. The Onsager corrections and state evolutions in these algorithms are defined by the free cumulants or rectangular free cumulants of the spectral distribution of W. Their forms were derived previously by Opper, Cakmak and Winther using nonrigorous dynamic functional theory techniques, and we provide rigorous proofs.Our motivating application is a Bayes-AMP algorithm for Principal Components Analysis, when there is prior structure for the principal components (PCs) and possibly nonwhite noise. For sufficiently large signal strengths and any non-Gaussian prior distributions for the PCs, we show that this algorithm provably achieves higher estimation accuracy than the sample PCs.