  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 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 MIMO relay system with correlated
source data and signal clipping, which can be modelled as a joint MMV-MT
system. While existing methods meet with difficulties in this example, OAMP
offers an efficient solution with excellent performance.
