Vector Approximate Message Passing With Arbitrary I.I.D. Noise Priors

TL;DR

The paper addresses sparse signal recovery from linear measurements $\boldsymbol{y}=\boldsymbol{A}\boldsymbol{x}+\boldsymbol{w}$ under non-Gaussian, i.i.d. noise by extending Vector AMP (VAMP) with a joint LMMSE estimation step that jointly infers $\boldsymbol{x}$ and $\boldsymbol{w}$. It leverages an expectation-propagation–like approximation to integrate noise priors $p_{\boldsymbol{w}}(\boldsymbol{w})$ into two MMSE denoisers and a linear MMSE module, producing updated mean and precision messages. Simulation results with Laplace and binary noise priors show significant reconstruction gains over standard VAMP, especially when the true noise deviates from Gaussian, demonstrating robustness to heavy-tailed and discrete noise. Overall, the work broadens the practical applicability of AMP-based inference to a wider class of non-Gaussian measurement-noise models in compressive sensing and related inverse problems.

Abstract

Approximate message passing (AMP) algorithms are devised under the Gaussianity assumption of the measurement noise vector. In this work, we relax this assumption within the vector AMP (VAMP) framework to arbitrary independent and identically distributed (i.i.d.) noise priors. We do so by rederiving the linear minimum mean square error (LMMSE) to accommodate both the noise and signal estimations within the message passing steps of VAMP. Numerical results demonstrate how our proposed algorithm handles non-Gaussian noise models as compared to VAMP. This extension to general noise priors enables the use of AMP algorithms in a wider range of engineering applications where non-Gaussian noise models are more appropriate.

Figures (4)

• Figure 1: Factor graph of the VAMP algorithm for arbitrary i.i.d. noise priors. Circles represent variable nodes and squares represent factor nodes.
• Figure 2: Block diagram of VAMP for arbitrary i.i.d. noise priors with its three modules: two denoising MMSE modules incorporating the prior information, $p_{\bm{\mathsf{x}}}(\cdot)$ and $p_{\bm{\mathsf{w}}}(\cdot)$, and the LMMSE module. The three modules exchange extrinsic information/messages through the blocks. The color of each module matches the color of the corresponding line numbers in Algorithm \ref{['algo:algorithm2']}.
• Figure 3: NRMSE of VAMP with arbitrary i.i.d. priors vs. standard VAMP as a function of the SNR with the noise vector drawn from the Laplace distribution with $\mu=0$.
• Figure 4: NRMSE of VAMP with arbitrary i.i.d. priors vs. standard VAMP as a function of the SNR with the noise vector drawn from the binary distribution.