A Mixture-Based Framework for Guiding Diffusion Models

Denoising diffusion models have driven significant progress in the field of Bayesian inverse problems. Recent approaches use pre-trained diffusion models as priors to solve a wide range of such problems, only leveraging inference-time compute and thereby eliminating the need to retrain task-specific models on the same dataset. To approximate the posterior of a Bayesian inverse problem, a diffusion model samples from a sequence of intermediate posterior distributions, each with an intractable likelihood function. This work proposes a novel mixture approximation of these intermediate distributions. Since direct gradient-based sampling of these mixtures is infeasible due to intractable terms, we propose a practical method based on Gibbs sampling. We validate our approach through extensive experiments on image inverse problems, utilizing both pixel- and latent-space diffusion priors, as well as on source separation with an audio diffusion model. The code is available at https://www.github.com/badr-moufad/mgdm

Variational Diffusion Posterior Sampling with Midpoint Guidance

Diffusion models have recently shown considerable potential in solving Bayesian inverse problems when used as priors. However, sampling from the resulting denoising posterior distributions remains a challenge as it involves intractable terms. To tackle this issue, state-of-the-art approaches formulate the problem as that of sampling from a surrogate diffusion model targeting the posterior and decompose its scores into two terms: the prior score and an intractable guidance term. While the former is replaced by the pre-trained score of the considered diffusion model, the guidance term has to be estimated. In this paper, we propose a novel approach that utilises a decomposition of the transitions which, in contrast to previous methods, allows a trade-off between the complexity of the intractable guidance term and that of the prior transitions. We validate the proposed approach through extensive experiments on linear and nonlinear inverse problems, including challenging cases with latent diffusion models as priors, and demonstrate its effectiveness in reconstructing electrocardiogram (ECG) from partial measurements for accurate cardiac diagnosis.

Divide-and-Conquer Posterior Sampling for Denoising Diffusion Priors

Interest in the use of Denoising Diffusion Models (DDM) as priors for solving inverse Bayesian problems has recently increased significantly. However, sampling from the resulting posterior distribution poses a challenge. To solve this problem, previous works have proposed approximations to bias the drift term of the diffusion. In this work, we take a different approach and utilize the specific structure of the DDM prior to define a set of intermediate and simpler posterior sampling problems, resulting in a lower approximation error compared to previous methods. We empirically demonstrate the reconstruction capability of our method for general linear inverse problems using synthetic examples and various image restoration tasks.

Invertible Flow Non Equilibrium sampling

Simultaneously sampling from a complex distribution with intractable normalizing constant and approximating expectations under this distribution is a notoriously challenging problem. We introduce a novel scheme, Invertible Flow Non Equilibrium Sampling (InFine), which departs from classical Sequential Monte Carlo (SMC) and Markov chain Monte Carlo (MCMC) approaches. InFine constructs unbiased estimators of expectations and in particular of normalizing constants by combining the orbits of a deterministic transform started from random initializations.When this transform is chosen as an appropriate integrator of a conformal Hamiltonian system, these orbits are optimization paths. InFine is also naturally suited to design new MCMC sampling schemes by selecting samples on the optimization paths.Additionally, InFine can be used to construct an Evidence Lower Bound (ELBO) leading to a new class of Variational AutoEncoders (VAE).