# Connecting Hamilton--Jacobi partial differential equations with maximum a posteriori and posterior mean estimators for some non-convex priors

arXiv: https://arxiv.org/abs/2104.11285 · HTML: https://mlsanity.com/paper/2104.11285/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 2 of its 69 references that are in the map.
- First posted on arXiv 22 Apr 2021.

## Builds on (most cited first, 2 of 2)

- [On Decomposition Models in Imaging Sciences and Multi-time Hamilton-Jacobi Partial Differential Equations](https://mlsanity.com/paper/1906.09502/) (2019)
- [On bayesian estimation and proximity operators](https://mlsanity.com/paper/1807.04021/) (2018)

## Built on by (most cited first, 3 of 3)

- [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)

## Abstract (arXiv)

Many imaging problems can be formulated as inverse problems expressed as finite-dimensional optimization problems. These optimization problems generally consist of minimizing the sum of a data fidelity and regularization terms. In [23,26], connections between these optimization problems and (multi-time) Hamilton--Jacobi partial differential equations have been proposed under the convexity assumptions of both the data fidelity and regularization terms. In particular, under these convexity assumptions, some representation formulas for a minimizer can be obtained. From a Bayesian perspective, such a minimizer can be seen as a maximum a posteriori estimator. In this chapter, we consider a certain class of non-convex regularizations and show that similar representation formulas for the minimizer can also be obtained. This is achieved by leveraging min-plus algebra techniques that have been originally developed for solving certain Hamilton--Jacobi partial differential equations arising in optimal control. Note that connections between viscous Hamilton--Jacobi partial differential equations and Bayesian posterior mean estimators with Gaussian data fidelity terms and log-concave priors have been highlighted in [25]. We also present similar results for certain Bayesian posterior mean estimators with Gaussian data fidelity and certain non-log-concave priors using an analogue of min-plus algebra techniques.