An Orthogonal Approximate Message Passing Framework for Multiuser Communications

We solve the open problem of constructing a Bayes-optimal iterative signal recovery algorithm for linear-Gaussian \emph{multiuser} communication systems with random precoding at the transmitters. Specifically, we consider the received signal model $\mathbf{y} = \sum_{u} \mathbf{H}_u \mathbf{\Xi}_u \mathbf{s}_u + \mathbf{n}$, where $\mathbf{n}$ is white Gaussian noise, $\{\mathbf{H}_u \in \mathbb{C}^{L \times L}\}$ are discrete-time channel matrices -- modeling a wide class of generally time-varying and dispersive linear channels with possibly multiple antennas -- and the precoding matrices $\{\boldsymbol{\Xi}_u \in \mathbb{C}^{L \times N_u}\}$ are drawn independently from a right-unitarily invariant random matrix ensemble. We consider generic \emph{non-separable} (coded) systems where the users' signals $\{\mathbf{s}_u\}$ follow general (non-factorizing) distributions. For this model, we introduce a novel orthogonal/vector approximate message passing (OAMP/VAMP)-type framework, including an algorithm and its high-dimensional (but finite-sample) analysis. From an algorithmic standpoint, the proposed method can be interpreted as an \emph{interpolation} between Minka's expectation propagation (EP)--a widely used method in machine learning--and OAMP. Our main theoretical contribution is the explicit finite-sample analysis of the proposed algorithm. Furthermore, we analyze the associated inference problem via a replica-symmetric (RS) ansatz by using a novel disorder-averaging technique. Both the (rigorous) high-dimensional analysis of the algorithm and the RS ansatz reveal the same decoupling principle, establishing that the proposed algorithm is asymptotically Bayes-optimal under the validity of the RS ansatz.