  This paper proposes a data-driven approach for constructing firmly
nonexpansive operators. We demonstrate its applicability in Plug-and-Play
methods, where classical algorithms such as forward-backward splitting,
Chambolle--Pock primal-dual iteration, Douglas--Rachford iteration or
alternating directions method of multipliers (ADMM), are modified by replacing
one proximal map by a learned firmly nonexpansive operator. We provide sound
mathematical background to the problem of learning such an operator via
expected and empirical risk minimization. We prove that, as the number of
training points increases, the empirical risk minimization problem converges
(in the sense of Gamma-convergence) to the expected risk minimization problem.
Further, we derive a solution strategy that ensures firmly nonexpansive and
piecewise affine operators within the convex envelope of the training set. We
show that this operator converges to the best empirical solution as the number
of points in the envelope increases in an appropriate sense. Finally, the
experimental section details practical implementations of the method and
presents an application in image denoising.
