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, \c{C}akmak, 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.
翻译:近似信件传递( AMP) 算法在各种应用中被广泛使用。 然而, Onsseger 校正和状态演进的确切形式取决于基本随机矩阵组合的特性,限制了白噪音产生的AMP算法可能适用于实践中产生的数据矩阵的程度。在这项工作中,我们研究更普遍的AMP算法,用于满足或垂直旋转法律差异的随机矩阵$W$W$,其中W$可能具有与白色噪音特征的半圆轴和马肯科-帕斯图法不同的光谱分布。Onsager 校正和状态演进取决于这些算法的特性,这些算法是由美元光谱分布的免费保流或矩形自由保量加以界定的。其形式先前由Oper、\c{C}akmak,以及Winther 使用非硬性动态功能理论技术,我们提供了严格的证据。 我们的激励应用程序是主要构件分析的Bayes-AMP算法, 其前结构是用于主要构件分析的精度, 之前的精确性结构是用于前方根基压的精确度的精确度。