Reconstruction of functions from prescribed proximal points

A Variational Inequality Model for the Construction of Signals from Inconsistent Nonlinear Equations

“…We show that firmly nonexpansive operators model many useful nonlinearities in this context. Key examples based on those of [27] are recalled and new ones are proposed. Following [27], we call p ∈ G a proximal point of y ∈ G relative to a firmly nonexpansive operator F : G → G if F y = p.…”

“…which is also the projection onto the closed convex set D = × 1 j m D j , is the hard clipper of [27,Example 2.11]. If we specialize to the case when, for every j ∈ {1, .…”

“…In [26,27], it was shown that many nonlinear observation processes found in signal processing, machine learning, and inference problems can be represented through such operators. This prompts us to consider the following formulation, whereby the prescriptions are modeled via Wiener systems (see Figure 1).…”

“…The work of [26,27] assumes that the prescription equations in Problem 1.1 are exact and hence that a solution exists. In many instances, however, the prescription operators may be imperfectly known or the model may be corrupted by perturbations, so that Problem 1.1 may not have solutions, e.g., [17,18,31].…”

“…In the proposed model, F i is a firmly nonexpansive operator. [26,27] are known to diverge in such situations. To deal robustly with possibly inconsistent equations, one must therefore come up with an appropriate relaxed formulation of Problem 1.1, i.e., one that seeks a point in C that satisfies the nonlinear equations in an approximate sense, and coincides with the original problem (1.4) if it happens to be consistent.…”