On Orthogonal Approximate Message Passing

Approximate Message Passing (AMP) is an efficient iterative parameter-estimation technique for certain high-dimensional linear systems with non-Gaussian distributions, such as sparse systems. In AMP, a so-called Onsager term is added to keep estimation errors approximately Gaussian. Orthogonal AMP (OAMP) does not require this Onsager term, relying instead on an orthogonalization procedure to keep the current errors uncorrelated with (i.e., orthogonal to) past errors. In this paper, we show that the orthogonality in OAMP ensures that errors are "asymptotically independently and identically distributed Gaussian" (AIIDG). This AIIDG property, which is essential for the attractive performance of OAMP, holds for separable functions. We present a procedure to realize the required orthogonality for OAMP through Gram-Schmidt orthogonalization (GSO). We show that expectation propagation (EP), AMP, OAMP and some other algorithms can be unified under this orthogonality framework. The simplicity and generality of OAMP provide efficient solutions for estimation problems beyond the classical linear models; related applications will be discussed in a companion paper where new algorithms are developed for problems with multiple constraints and multiple measurement variables.