  Inverse problems consist of recovering a signal from a collection of noisy
measurements. These are typically cast as optimization problems, with classic
approaches using a data fidelity term and an analytic regularizer that
stabilizes recovery. Recent Plug-and-Play (PnP) works propose replacing the
operator for analytic regularization in optimization methods by a data-driven
denoiser. These schemes obtain state of the art results, but at the cost of
limited theoretical guarantees. To bridge this gap, we present a new algorithm
that takes samples from the manifold of true data as input and outputs an
approximation of the projection operator onto this manifold. Under standard
assumptions, we prove this algorithm generates a learned operator, called
Wasserstein-based projection (WP), that approximates the true projection with
high probability. Thus, WPs can be inserted into optimization methods in the
same manner as PnP, but now with theoretical guarantees. Provided numerical
examples show WPs obtain state of the art results for unsupervised PnP signal
recovery.
