What Is the Motivation of Proximal Mapping / Proximal Operator?

Question

For a convex function $h$, its proximal operator is defined as: $$\operatorname{prox}_h(x)=\arg\min_u \Big(h(u)+\frac{1}{2}\|u-x\|^2\Big)$$ Can anyone provide an intuitive explanation/motivation of proximal mapping?

Answer 1

There are many good answers here already, but for the sake of completeness I will add on more intuition. The proximal operator of $g$ (with parameter $\lambda$) can also be seen as a gradient step (with stepsize $\lambda$) with respect to the Moreau envelope $g_\lambda$ of $g$. The Moreau envelope is a smooth under approximation of the original function and is given by $$ g_\lambda(x) = \min_u \{g(u) + \frac{1}{2 \lambda} || u-x||^2\}. $$ Evidently its definition is closely related to the proximal operator. The above mentioned identity then reads as $$ \operatorname{prox}_{\lambda g}(x) = x - \lambda \nabla g_\lambda(x). $$