Summary of Hj-sampler: a Bayesian Sampler For Inverse Problems Of a Stochastic Process by Leveraging Hamilton-jacobi Pdes and Score-based Generative Models, By Tingwei Meng et al.
HJ-sampler: A Bayesian sampler for inverse problems of a stochastic process by leveraging Hamilton-Jacobi PDEs and score-based generative models
by Tingwei Meng, Zongren Zou, Jérôme Darbon, George Em Karniadakis
First submitted to arxiv on: 15 Sep 2024
Categories
- Main: Machine Learning (cs.LG)
- Secondary: Optimization and Control (math.OC); Computation (stat.CO)
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| Summary difficulty | Written by | Summary |
|---|---|---|
| High | Paper authors | High Difficulty Summary Read the original abstract here |
| Medium | GrooveSquid.com (original content) | Medium Difficulty Summary This paper explores the interplay between stochastic processes and optimal control in the context of diffusion models. It builds on the log transform, known as the Cole-Hopf transform, and extends it within a more abstract framework that includes a linear operator. The authors demonstrate how this framework relates to Bayesian inference under specific initial and terminal conditions. They propose a new algorithm called the HJ-sampler for solving Bayesian inverse problems involving stochastic differential equations with given terminal observations. The algorithm involves two stages: solving viscous Hamilton-Jacobi partial differential equations and sampling from the associated stochastic optimal control problem. The authors introduce two variants of the solver, the Riccati-HJ-sampler and the SGM-HJ-sampler, which utilize different methods to solve the PDEs. They demonstrate the effectiveness and flexibility of their proposed methods by applying them to various stochastic processes and prior distributions. |
| Low | GrooveSquid.com (original content) | Low Difficulty Summary This paper is about using math to understand how things move around in a way that’s not totally predictable. It uses a special tool called the Cole-Hopf transform to figure out how things are connected. The authors come up with a new method for solving problems where we don’t know what happened in the past, but we do know what happened at the end. This method is good because it lets us try different ways of solving the problem and see which one works best. |
Keywords
» Artificial intelligence » Bayesian inference » Diffusion
