A Hamilton-Jacobi-based Proximal Operator

Stanley Osher, Samy Wu Fung, Howard Heaton

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 only known for 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 importance 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 only accessible by (possibly noisy) blackbox samples. We show HJ-Prox is effective numerically via several examples.

Publication

A Hamilton-Jacobi-based proximal operator (arXiv Link).

@article{osher2023hamilton,
         title={{A Hamilton-Jacobi-based proximal operator}},
         author={Osher, Stanley and Heaton, Howard and Fung, Samy Wu},
         journal={{Proceedings of the National Academy of Sciences}},
         year={2023},
         volume={120},
         number={14}
}

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