  This paper introduces a framework for approximate message passing (AMP) in
dynamic settings where the data at each iteration is passed through a linear
operator. This framework is motivated in part by applications in large-scale,
distributed computing where only a subset of the data is available at each
iteration. An autoregressive memory term is used to mitigate information loss
across iterations and a specialized algorithm, called projection AMP, is
designed for the case where each linear operator is an orthogonal projection.
Precise theoretical guarantees are provided for a class of Gaussian matrices
and non-separable denoising functions. Specifically, it is shown that the
iterates can be well-approximated in the high-dimensional limit by a Gaussian
process whose second-order statistics are defined recursively via state
evolution. These results are applied to the problem of estimating a rank-one
spike corrupted by additive Gaussian noise using partial row updates, and the
theory is validated by numerical simulations.
