论文标题
近似消息传递旋转矩阵的算法
Approximate Message Passing algorithms for rotationally invariant matrices
论文作者
论文摘要
近似消息传递（AMP）算法已经在各种应用程序中广泛使用。但是，其onsager校正和状态演变的精确形式取决于基础随机矩阵集合的特性，从而限制了用于白噪声的AMP算法的程度，可能适用于实践中出现的数据矩阵。 在这项工作中，我们研究了满足法律正交旋转不变性的随机矩阵$ W $的更通用的AMP算法，在该法律上，$ w $的光谱分布可能与半圆和Marcenko-Pastur法律不同。这些算法中的Onsager校正和状态演变由$ W $的光谱分布的自由累积或矩形无累积累积物来定义。它们的形式先前是由OPPER，çakmak和Winther得出的，并使用非鲁and动态功能理论技术得出，我们提供了严格的证据。 当主成分（PC）和可能的非白噪声存在先验结构时，我们激励的应用是用于主组件分析的贝叶斯 - AMP算法。对于足够较大的信号强度和PC的任何非高斯先前分布，我们表明该算法比样品PC可以实现更高的估计精度。
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.
