Approximate Message Passing with A Class of Non-Separable Denoisers

Approximate message passing (AMP) is a class of low-complexity scalable algorithms for solving high-dimensional linear regression tasks where one wishes to recover an unknown signal β0 from noisy, linear measurements y = Aβ0 + w. AMP has the attractive feature that its performance (for example, the mean squared error of its estimates) can be accurately tracked by a simple, scalar iteration referred to as state evolution when the unknown signal has independent and identically distributed (i.i.d.) entries. However, in many real-world applications, like image or audio signal reconstruction, the unknown signal contains dependencies between entries and so a coordinate-wise independence structure is not a good approximation for the prior of the unknown signal. In this paper we study the case where the unknown signal has dependent entries using a class of non-separable sliding-window denoisers and prove that a new form of state evolution still accurately predicts AMP performance in this scenario. This is an early step in understanding the role of non-separable denoisers within AMP, and will lead to a characterization of more general denoisers in problems including compressive image reconstruction.