旋转不变矩阵的近似消息传递算法
Approximate Message Passing algorithms for rotationally invariant matrices
摘要：
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集合的性质，限制了针对白噪声导出的AMP算法可适用于实践中出现的数据矩阵的程度。在这项工作中，我们研究了更一般的AMP算法随机矩阵$W$，满足正交旋转不变性的法律，其中$W$可能有一个谱分布，是不同的白噪声的特性的周期和Marcenko-Pastur法律。这些算法中的Onsager修正和状态演化由$W$谱分布的自由累积量或矩形自由累积量定义。它们的形式是以前由Opper，Jakmak和Winther使用非严格的动态泛函理论技术推导出来的，我们提供了严格的证明。我们的激励应用程序是主成分分析的贝叶斯-AMP算法，当主成分（PC）和可能的非白噪声的先验结构。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。
英文摘要：
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, Çakmak, and Winther using non-rigorous 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 non-white 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.
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