An Expectation-Maximization Approach to Tuning Generalized Vector Approximate Message Passing

Generalized Vector Approximate Message Passing (GVAMP) is an efficient iterative algorithm for approximately minimum-mean-squared-error estimation of a random vector $\mathbf{x}\sim p_{\mathbf{x}}(\mathbf{x})$ from generalized linear measurements, i.e., measurements of the form $\mathbf{y}=Q(\mathbf{z})$ where $\mathbf{z}=\mathbf{Ax}$ with known $\mathbf{A}$, and $Q(\cdot)$ is a noisy, potentially nonlinear, componentwise function. Problems of this form show up in numerous applications, including robust regression, binary classification, quantized compressive sensing, and phase retrieval. In some cases, the prior $p_{\mathbf{x}}$ and/or channel $Q(\cdot)$ depend on unknown deterministic parameters $\boldsymbol{\theta}$, which prevents a direct application of GVAMP. In this paper we propose a way to combine expectation maximization (EM) with GVAMP to jointly estimate $\mathbf{x}$ and $\boldsymbol{\theta}$. We then demonstrate how EM-GVAMP can solve the phase retrieval problem with unknown measurement-noise variance

Expectation-Maximization-Aided Hybrid Generalized Expectation Consistent for Sparse Signal Reconstruction

The reconstruction of sparse signal is an active area of research. Different from a typical i.i.d. assumption, this paper considers a non-independent prior of group structure. For this more practical setup, we propose EM-aided HyGEC, a new algorithm to address the stability issue and the hyper-parameter issue facing the other algorithms. The instability problem results from the ill condition of the transform matrix, while the unavailability of the hyper-parameters is a ground truth that their values are not known beforehand. The proposed algorithm is built on the paradigm of HyGAMP (proposed by Rangan et al.) but we replace its inner engine, the GAMP, by a matrix-insensitive alternative, the GEC, so that the first issue is solved. For the second issue, we take expectation-maximization as an outer loop, and together with the inner engine HyGEC, we learn the value of the hyper-parameters. Effectiveness of the proposed algorithm is also verified by means of numerical simulations.Comment: 15 pages, 4 figures. This paper has been submitted to IEEE Signal Processing Letter