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

Approximate message passing (AMP) algorithms break a (high-dimensional) statistical problem into manageable parts and then repeatedly solve each part in turn, akin to alternating projections. One notable characteristic is that their asymptotic behaviours can be accurately predicted via their associated 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 (MMVs) and multiple transforms (MTs). We name the resulting algorithms as 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 orthogonal principle underpinning OAMP makes these extensions straightforward. The MMV and MT models are widely applicable to signal processing and communications. We present an example of a MIMO relay system with correlated source data and signal clipping, which can be effectively modelled as a joint MMV-MT system. While existing methods encounter challenges when applied to this example, OAMP offers an efficient solution with excellent performance.