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Approximate message passing (AMP) algorithms break a (high-dimensional)statistical problem into parts then repeatedly solve each part in turn, akin toalternating projections. A distinguishing feature is their asymptoticbehaviours can be accurately predicted via their associated state evolutionequations. Orthogonal AMP (OAMP) was recently developed to avoid the need forcomputing the so-called Onsager term in traditional AMP algorithms, providingtwo clear benefits: the derivation of an OAMP algorithm is both straightforwardand more broadly applicable. OAMP was originally demonstrated for statisticalproblems with a single measurement vector and single transform. This paperextends OAMP to statistical problems with multiple measurement vectors (MMVs)and multiple transforms (MTs). We name the resulting algorithms as OAMP-MMV andOAMP-MT respectively, and their combination as augmented OAMP (A-OAMP). Whereasthe extension of traditional AMP algorithms to such problems would bechallenging, the orthogonal principle underpinning OAMP makes these extensionsstraightforward.  The MMV and MT models are widely applicable to signal processing andcommunications. We present an example of MIMO relay system with correlatedsource data and signal clipping, which can be modelled as a joint MMV-MTsystem. While existing methods meet with difficulties in this example, OAMPoffers an efficient solution with excellent performance.

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