Regularized Approximate Message Passing for Overloaded Discrete Linear Inversion

We propose regularized approximate message passing (RAMP), a low-complexity algorithm for discrete signal detection in overloaded multiple-input multiple-output (MIMO) systems where the number of transmit antennas exceeds the number of receive antennas. While the state-of-the-art (SotA) iterative discrete least squares (IDLS) framework achieves near-optimal discrete-aware performance, its iterative matrix inversions impose a prohibitive $\mathcal{O}(M^3)$ complexity. RAMP resolves this by deriving an adaptive, state-dependent scalar denoiser that enforces arbitrary discrete constellation constraints within the approximate message passing (AMP) framework, reducing per-iteration complexity to $\mathcal{O}(NM)$. A robust variant is further proposed by incorporating an $\ell_2$-norm penalty, analogous to a linear minimum mean squared error (LMMSE) estimator, to enhance noise resilience. Simulation results under uncorrelated Rayleigh fading demonstrate that both proposed algorithms closely track their exact IDLS counterparts while avoiding the catastrophic failure of standard AMP in the overloaded regime, achieving steep bit error rate (BER) waterfall curves at a fraction of the computational cost.