HJ-sampler: A Bayesian sampler for inverse problems of a stochastic process by leveraging Hamilton-Jacobi PDEs and score-based generative models

• URL: http://arxiv.org/abs/2409.09614v2
• Date: Tue, 8 Oct 2024 20:50:43 GMT
• Title: HJ-sampler: A Bayesian sampler for inverse problems of a stochastic process by leveraging Hamilton-Jacobi PDEs and score-based generative models
• Authors: Tingwei Meng, Zongren Zou, Jérôme Darbon, George Em Karniadakis,
• Abstract summary: This paper builds on the log transform known as the Cole-Hopf transform in Brownian motion contexts. We develop a new algorithm, named the HJ-sampler, for inference for the inverse problem of a differential equation with given terminal observations.
• Score: 1.949927790632678
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Abstract: The interplay between stochastic processes and optimal control has been extensively explored in the literature. With the recent surge in the use of diffusion models, stochastic processes have increasingly been applied to sample generation. This paper builds on the log transform, known as the Cole-Hopf transform in Brownian motion contexts, and extends it within a more abstract framework that includes a linear operator. Within this framework, we found that the well-known relationship between the Cole-Hopf transform and optimal transport is a particular instance where the linear operator acts as the infinitesimal generator of a stochastic process. We also introduce a novel scenario where the linear operator is the adjoint of the generator, linking to Bayesian inference under specific initial and terminal conditions. Leveraging this theoretical foundation, we develop a new algorithm, named the HJ-sampler, for Bayesian inference for the inverse problem of a stochastic differential equation with given terminal observations. The HJ-sampler involves two stages: (1) solving the viscous Hamilton-Jacobi partial differential equations, and (2) sampling from the associated stochastic optimal control problem. Our proposed algorithm naturally allows for flexibility in selecting the numerical solver for viscous HJ PDEs. We introduce two variants of the solver: the Riccati-HJ-sampler, based on the Riccati method, and the SGM-HJ-sampler, which utilizes diffusion models. We demonstrate the effectiveness and flexibility of the proposed methods by applying them to solve Bayesian inverse problems involving various stochastic processes and prior distributions, including applications that address model misspecifications and quantifying model uncertainty.

Related papers

• A Stein Gradient Descent Approach for Doubly Intractable Distributions [5.63014864822787]
  We propose a novel Monte Carlo Stein variational gradient descent (MC-SVGD) approach for inference for doubly intractable distributions. The proposed method achieves substantial computational gains over existing algorithms, while providing comparable inferential performance for the posterior distributions.
  arXiv  Detail & Related papers  (2024-10-28T13:42:27Z)
• Unified Convergence Analysis for Score-Based Diffusion Models with Deterministic Samplers [49.1574468325115]
  We introduce a unified convergence analysis framework for deterministic samplers. Our framework achieves iteration complexity of $tilde O(d2/epsilon)$. We also provide a detailed analysis of Denoising Implicit Diffusion Models (DDIM)-type samplers.
  arXiv  Detail & Related papers  (2024-10-18T07:37:36Z)
• Stochastic Sampling from Deterministic Flow Models [8.849981177332594]
  We present a method to turn flow models into a family of differential equations (SDEs) that have the same marginal distributions. We empirically demonstrate advantages of our method on a toy Gaussian setup and on the large scale ImageNet generation task.
  arXiv  Detail & Related papers  (2024-10-03T05:18:28Z)
• Harmonic Path Integral Diffusion [0.4527270266697462]
  We present a novel approach for sampling from a continuous multivariate probability distribution, which may either be explicitly known (up to a normalization factor) or represented via empirical samples. Our method constructs a time-dependent bridge from a delta function centered at the origin of the state space at $t=0$, transforming it into the target distribution at $t=1$. We contrast these algorithms with other sampling methods, particularly simulated and path integral sampling, highlighting their advantages in terms of analytical control, accuracy, and computational efficiency.
  arXiv  Detail & Related papers  (2024-09-23T16:20:21Z)
• Total Uncertainty Quantification in Inverse PDE Solutions Obtained with Reduced-Order Deep Learning Surrogate Models [50.90868087591973]
  We propose an approximate Bayesian method for quantifying the total uncertainty in inverse PDE solutions obtained with machine learning surrogate models. We test the proposed framework by comparing it with the iterative ensemble smoother and deep ensembling methods for a non-linear diffusion equation.
  arXiv  Detail & Related papers  (2024-08-20T19:06:02Z)
• Weak Generative Sampler to Efficiently Sample Invariant Distribution of Stochastic Differential Equation [8.67581853745823]
  Current deep learning-based method solves the stationary Fokker--Planck equation to determine the invariant probability density function in form of deep neural networks. We introduce a framework that employs a weak generative sampler (WGS) to directly generate independent and identically distributed (iid) samples. Our proposed loss function is based on the weak form of the Fokker--Planck equation, integrating normalizing flows to characterize the invariant distribution.
  arXiv  Detail & Related papers  (2024-05-29T16:41:42Z)
• Gaussian Mixture Solvers for Diffusion Models [84.83349474361204]
  We introduce a novel class of SDE-based solvers called GMS for diffusion models. Our solver outperforms numerous SDE-based solvers in terms of sample quality in image generation and stroke-based synthesis.
  arXiv  Detail & Related papers  (2023-11-02T02:05:38Z)
• Heterogeneous Multi-Task Gaussian Cox Processes [61.67344039414193]
  We present a novel extension of multi-task Gaussian Cox processes for modeling heterogeneous correlated tasks jointly. A MOGP prior over the parameters of the dedicated likelihoods for classification, regression and point process tasks can facilitate sharing of information between heterogeneous tasks. We derive a mean-field approximation to realize closed-form iterative updates for estimating model parameters.
  arXiv  Detail & Related papers  (2023-08-29T15:01:01Z)
• A Geometric Perspective on Diffusion Models [57.27857591493788]
  We inspect the ODE-based sampling of a popular variance-exploding SDE. We establish a theoretical relationship between the optimal ODE-based sampling and the classic mean-shift (mode-seeking) algorithm.
  arXiv  Detail & Related papers  (2023-05-31T15:33:16Z)
• Plug-and-Play split Gibbs sampler: embedding deep generative priors in Bayesian inference [12.91637880428221]
  This paper introduces a plug-and-play sampling algorithm that leverages variable splitting to efficiently sample from a posterior distribution. It divides the challenging task of posterior sampling into two simpler sampling problems. Its performance is compared to recent state-of-the-art optimization and sampling methods.
  arXiv  Detail & Related papers  (2023-04-21T17:17:51Z)
• Restoration-Degradation Beyond Linear Diffusions: A Non-Asymptotic Analysis For DDIM-Type Samplers [90.45898746733397]
  We develop a framework for non-asymptotic analysis of deterministic samplers used for diffusion generative modeling. We show that one step along the probability flow ODE can be expressed as two steps: 1) a restoration step that runs ascent on the conditional log-likelihood at some infinitesimally previous time, and 2) a degradation step that runs the forward process using noise pointing back towards the current gradient.
  arXiv  Detail & Related papers  (2023-03-06T18:59:19Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.