  Vector approximate message passing (VAMP) is a computationally simple
approach to the recovery of a signal $\mathbf{x}$ from noisy linear
measurements $\mathbf{y}=\mathbf{Ax}+\mathbf{w}$. Like the AMP proposed by
Donoho, Maleki, and Montanari in 2009, VAMP is characterized by a rigorous
state evolution (SE) that holds under certain large random matrices and that
matches the replica prediction of optimality. But while AMP's SE holds only for
large i.i.d. sub-Gaussian $\mathbf{A}$, VAMP's SE holds under the much larger
class: right-rotationally invariant $\mathbf{A}$. To run VAMP, however, one
must specify the statistical parameters of the signal and noise. This work
combines VAMP with Expectation-Maximization to yield an algorithm, EM-VAMP,
that can jointly recover $\mathbf{x}$ while learning those statistical
parameters. The fixed points of the proposed EM-VAMP algorithm are shown to be
stationary points of a certain constrained free-energy, providing a variational
interpretation of the algorithm. Numerical simulations show that EM-VAMP is
robust to highly ill-conditioned $\mathbf{A}$ with performance nearly matching
oracle-parameter VAMP.
