We study a class of Approximate Message Passing (AMP) algorithms for symmetric and rectangular spiked random matrix models with orthogonally invariant noise. The AMP iterates have fixed dimension $K \geq 1$, a multivariate non-linearity is applied in each AMP iteration, and the algorithm is spectrally initialized with $K$ super-critical sample eigenvectors. We derive the forms of the Onsager debiasing coefficients and corresponding AMP state evolution, which depend on the free cumulants of the noise spectral distribution. This extends previous results for such models with $K=1$ and an independent initialization. Applying this approach to Bayesian principal components analysis, we introduce a Bayes-OAMP algorithm that uses as its non-linearity the posterior mean conditional on all preceding AMP iterates. We describe a practical implementation of this algorithm, where all debiasing and state evolution parameters are estimated from the observed data, and we illustrate the accuracy and stability of this approach in simulations.

翻译：我们研究一种具有对称和矩形随机随机矩阵模型的近似电文传递算法(AMP)类,该算法具有恒定尺寸 $K\geq 1 美元,在每种AMP迭代中应用多变量非线性,该算法是光谱初始化的,使用$K$超临界样本源代数。我们得出了Onsager 降低偏差系数和相应的AMP 状态演进形式,这些形式取决于噪音光谱分布的自由累积值。这扩大了这些模型以前的结果,以$K=1美元和独立初始化方式计算出这些模型的结果。在对巴伊西亚主要部件进行分析时,我们采用了一种Bayes-OAMP算法作为非线性,将后端平均值作为此前所有AMP代数的不线性条件。我们描述了这一算法的实际实施情况,根据观察到的数据估算出所有偏差和状态演进参数,我们演示了模拟方法的准确性和稳定性。