The generalized linear model (GLM), where a random vector $\boldsymbol{ x}$ is observed through a noisy, possibly nonlinear, function of a linear transform output $\boldsymbol{ z}=\boldsymbol{ Ax}$, arises in a range of applications such as robust regression, binary classification, quantized compressed sensing, phase retrieval, photon-limited imaging, and inference from neural spike trains. When $\boldsymbol{ A}$ is large and i.i.d. Gaussian, the generalized approximate message passing (GAMP) algorithm is an efficient means of MAP or marginal inference, and its performance can be rigorously characterized by a scalar state evolution. For general $\boldsymbol{ A}$, though, GAMP can misbehave. Damping and sequential-updating help to robustify GAMP, but their effects are limited. Recently, a "vector AMP" (VAMP) algorithm was proposed for additive white Gaussian noise channels. VAMP extends AMP's guarantees from i.i.d. Gaussian $\boldsymbol{ A}$ to the larger class of rotationally invariant $\boldsymbol{ A}$. In this paper, we show how VAMP can be extended to the GLM. Numerical experiments show that the proposed GLM-VAMP is much more robust to ill-conditioning in $\boldsymbol{ A}$ than damped GAMP.
Categories: cs.IT math.IT