Approximate Message Passing algorithms for rotationally invariant matrices

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 underlying random matrix ensemble, limiting extent to which AMP derived white noise may be applicable data matrices that arise in practice. In this work, we study more general W satisfy orthogonal rotational invariance law, where spectral distribution is different from semicircle Marcenko–Pastur laws characteristic noise. The these are defined by free cumulants or rectangular W. Their were previously Opper, Çakmak Winther using nonrigorous dynamic functional theory techniques, provide rigorous proofs. Our motivating application Bayes-AMP algorithm Principal Components Analysis, when there prior structure principal components (PCs) possibly nonwhite For sufficiently large signal strengths any non-Gaussian distributions PCs, show provably achieves higher estimation accuracy than sample PCs.