TITLE: When is a mapping the proximity operator of some convex function?
QUESTION [9 upvotes]: Is there a characterization of mappings $p : \mathbb R^n \rightarrow \mathbb R^n$ which are proximity operators (in the sense of Moreau) of l.s.c (extended) real-valued functions ?
That is, given $p : \mathbb R^n \rightarrow \mathbb R^n$, under what sufficient conditions does there exist an extended-valued l.s.c  convex function $g:\mathbb R^n \rightarrow (-\infty, +\infty]$ such that
$$p(x) \equiv \mathrm{prox}_g(x) := \underset{z \in \mathbb R^n}{\text{argmin }}\frac{1}{2}\|z-x\|_2^2 + g(z) \;?$$
N.B: Of course it's necessary that $p$ be firmly-nonexpansive, and have other classical properties of prox operators.
Motivation: In regularization techniques for signal / image processing, one usually proposes to minimize an energy of the form $f(x) + g(x)$, where $x=x^*$ is the image to be recovered from noisy / corrupted measures, $f(x)$ is a data fidelity and measures the "fit" of the model, while $g(x)$ is a regularization term that imposes some structural constraints. For example, one can take $f(x) = \frac{1}{2}\|y-Ax\|_2^2$, under a additve Gaussian-noise assumption, where $y$ is the observed image and $A$ is a sensing linear operator, so that $y \approx Ax + \text{ noise}$, etc., etc.
A brilliant idea that has been proposed in Social Sparsity! is to impose the penalty $g$ only implicitly, by instead constructing its proximal operator $p(x)$, i.e by stating the intended shrinkage action of $g$ on the model coefficients $x_j$.
For a concrete example, think of a (fictional) world in which we didn't know about the $\ell_1$ norm, but instead decided to invent the Lasso by stating that the prox of the (unknown) $\ell_1$ penalty should shrink the coefficients according to the soft-thresholder
$$(p(x))_j = st_{\lambda}(x_j) = sign(x_j)(|x_j| - \lambda)_+,$$

where $\lambda > 0$ is a regularization parameter and $sign(x_j): = -sign(-x_j) = 1$ if $ x_j > 0$ and $0$ else. Note that the above prox would correspond to a penalty $g(x) = \lambda \|x\|_1$, and acts component-wise only because we're assuming (in this example) a separable penalty.
The question is then: How to show that $st_{\lambda}$ actually corresponds to the proximal operator of some penalty function.

REPLY [4 votes]: You may be interested in the characterization of proximity operators (of convex or nonconvex functions) provided in the paper https://hal.inria.fr/hal-01835101
In fact, the Moreau paper (Corollary 10.c) shows that $f: \mathcal{H} \to \mathcal{H}$ is the proximal map associated to some convex lsc $\varphi: \mathcal{H} \to \mathbb{R}$ if, and only if, the following two conditions hold: 

$f$ is the subdifferential of some convex lsc function $\psi: \mathcal{H} \to \mathbb{R} \cup \{+\infty\}$;
$f$ is non-expansive

Here $\mathcal{H}$ is any Hilbert space, finite or infinite dimensional.
To answer your question, one also needs to consider the case where $\varphi$ may be nonconvex. In https://hal.inria.fr/hal-01835101 it is shown (Theorem 1) that: $f: \mathcal{Y} \subset \mathcal{H} \to \mathcal{H}$ is the proximal map associated to some (possibly nonconvex) $\varphi: \mathcal{H} \to \mathbb{R}$ if, and only if, $f$ is the subdifferential of some convex lsc function $\psi : \mathcal{H} \to \mathbb{R} \cup \{+\infty\}$.
In practice, it may not always look straightforward to check that $f$ is a subdifferential (this is related to notions such as cyclic monotonicity). Luckily this is easier when $f$ is $C^1$ (Theorem 2): when $f$ is $C^1$, it is the proximity operator of some $\varphi$ if, and only if, its differential $Df(y)$ is symmetric positive semi-definite.
The case of social sparsity shrinkage operators is precisely considered in Section 1.4: using Theorem 2, it is shown (Corollary 5) that a family of social shrinkage operators are  not proximity operators, except in classical cases where they match group-sparsity shrinkage with disjoint groups.  
As far as I know, Moreau fully characterizes proximity operators of convex $\varphi$ are fully characterized 
by Moreau -a function f