A Hamilton–Jacobi-based proximal operator

Abstract

First-order optimization algorithms are widely used today. Two standard building blocks in these algorithms are proximal operators (proximals) and gradients. Although gradients can be computed for a wide array of functions, explicit proximal formulas are known for only limited classes of functions. We provide an algorithm, HJ-Prox, for accurately approximating such proximals. This is derived from a collection of relations between proximals, Moreau envelopes, Hamilton–Jacobi (HJ) equations, heat equations, and Monte Carlo sampling. In particular, HJ-Prox smoothly approximates the Moreau envelope and its gradient. The smoothness can be adjusted to act as a denoiser. Our approach applies even when functions are accessible only by (possibly noisy) black box samples. We show that HJ-Prox is effective numerically via several examples.

Optimised Sparse Image Reconstruction Via Densification and Relocation

“…For descent algorithms, our aim is to find nearly global optimal solutions using the Hamilton-Jacobi equation-based proximal operator (Heaton et al, 2023;Osher et al, 2023) [36,54].…”