Orthogonal AMP for Problems with Multiple Measurement Vectors and/or Multiple Transforms

Approximate message passing (AMP) algorithms break a (high-dimensional) statistical problem into parts then repeatedly solve each part in turn, akin to alternating projections. A distinguishing feature is their asymptotic behaviours can be accurately predicted via state evolution equations. Orthogonal AMP (OAMP) was recently developed to avoid the need for computing the so-called Onsager term in traditional AMP algorithms, providing two clear benefits: the derivation of an OAMP algorithm is both straightforward and more broadly applicable. OAMP was originally demonstrated for statistical problems with a single measurement vector and single transform. This paper extends OAMP to statistical problems with multiple measurement vectors (MMV) and multiple transforms (MT). The resulting algorithms are named OAMP-MMV and OAMP-MT respectively, and their combination as augmented OAMP (A-OAMP). Whereas the extension of traditional AMP algorithms to such problems would be challenging, the orthogonality principle underpinning OAMP makes these extensions straightforward. A full verison of this paper is accessible at arxiv (see [1]).