# A Hamilton-Jacobi-based Proximal Operator

arXiv: https://arxiv.org/abs/2211.12997 · HTML: https://mlsanity.com/paper/2211.12997/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Nov 2022.

## Built on by (most cited first, 13 of 13)

- [Wasserstein proximal operators describe score-based generative models and resolve memorization](https://mlsanity.com/paper/2402.06162/) (2024)
- [Flow-based Distributionally Robust Optimization](https://mlsanity.com/paper/2310.19253/) (2023)
- [Noise-Free Sampling Algorithms via Regularized Wasserstein Proximals](https://mlsanity.com/paper/2308.14945/) (2023)
- [Diffusion at Absolute Zero: Langevin Sampling Using Successive Moreau Envelopes [conference paper]](https://mlsanity.com/paper/2502.01358/) (2025)
- [HJ-sampler: A Bayesian sampler for inverse problems of a stochastic process by leveraging Hamilton-Jacobi PDEs and score-based generative models](https://mlsanity.com/paper/2409.09614/) (2024)
- [Consensus-based optimization for closed-box adversarial attacks and a connection to evolution strategies](https://mlsanity.com/paper/2506.24048/) (2025)
- [From exponential to finite/fixed-time stability: Applications to optimization](https://mlsanity.com/paper/2409.11713/) (2024)
- [Thinking Out of the Box: Hybrid SAT Solving by Unconstrained Continuous Optimization](https://mlsanity.com/paper/2506.00674/) (2025)
- [Deep learning methods for inverse problems using connections between proximal operators and Hamilton-Jacobi equations](https://mlsanity.com/paper/2512.23829/) (2025)
- [Probabilistic Gaussian Homotopy: A Probability-Space Continuation Framework for Nonconvex Optimization](https://mlsanity.com/paper/2603.13546/) (2026)
- [Implicit Neural Optimal Transport via Fixed-Point Optimization](https://mlsanity.com/paper/2605.10792/) (2026)
- [Viscosity-Informed Generative Actor-Critic for High-Dimensional Stochastic Optimal Control](https://mlsanity.com/paper/2605.15668/) (2026)
- [Proximal basin hopping: global optimization with guarantees](https://mlsanity.com/paper/2605.18364/) (2026)

## Abstract (arXiv)

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