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.
Categories: cs.IT math.IT