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Divide-and-Conquer Posterior Sampling for Denoising Diffusion Priors

Yazid Janati, Badr Moufad, Alain Durmus, Eric Moulines, Jimmy Olsson

TL;DR

This work presents an innovative framework, divide-and-conquer posterior sampling, which leverages the inherent structure of DDMs to construct a sequence of intermediate posteriors that guide the produced samples to the target posterior.

Abstract

Recent advancements in solving Bayesian inverse problems have spotlighted denoising diffusion models (DDMs) as effective priors. Although these have great potential, DDM priors yield complex posterior distributions that are challenging to sample. Existing approaches to posterior sampling in this context address this problem either by retraining model-specific components, leading to stiff and cumbersome methods, or by introducing approximations with uncontrolled errors that affect the accuracy of the produced samples. We present an innovative framework, divide-and-conquer posterior sampling, which leverages the inherent structure of DDMs to construct a sequence of intermediate posteriors that guide the produced samples to the target posterior. Our method significantly reduces the approximation error associated with current techniques without the need for retraining. We demonstrate the versatility and effectiveness of our approach for a wide range of Bayesian inverse problems. The code is available at \url{https://github.com/Badr-MOUFAD/dcps}

Divide-and-Conquer Posterior Sampling for Denoising Diffusion Priors

TL;DR

This work presents an innovative framework, divide-and-conquer posterior sampling, which leverages the inherent structure of DDMs to construct a sequence of intermediate posteriors that guide the produced samples to the target posterior.

Abstract

Recent advancements in solving Bayesian inverse problems have spotlighted denoising diffusion models (DDMs) as effective priors. Although these have great potential, DDM priors yield complex posterior distributions that are challenging to sample. Existing approaches to posterior sampling in this context address this problem either by retraining model-specific components, leading to stiff and cumbersome methods, or by introducing approximations with uncontrolled errors that affect the accuracy of the produced samples. We present an innovative framework, divide-and-conquer posterior sampling, which leverages the inherent structure of DDMs to construct a sequence of intermediate posteriors that guide the produced samples to the target posterior. Our method significantly reduces the approximation error associated with current techniques without the need for retraining. We demonstrate the versatility and effectiveness of our approach for a wide range of Bayesian inverse problems. The code is available at \url{https://github.com/Badr-MOUFAD/dcps}
Paper Structure (51 sections, 2 theorems, 49 equations, 15 figures, 7 tables, 1 algorithm)

This paper contains 51 sections, 2 theorems, 49 equations, 15 figures, 7 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notation.
  3. Posterior sampling with DDM prior
  4. DDM priors.
  5. Posterior sampling.
  6. The DCPS algorithm
  7. Smoothing the DPS approximation.
  8. Intermediate posteriors.
  9. Sampling the initial distribution.
  10. Sampling the transitions.
  11. Intermediate potentials.
  12. Experiments
  13. Gaussian mixture.
  14. Imaging experiment.
  15. Trajectory prediction.
  16. ...and 36 more sections

Key Result

Proposition 1

Let $k \in \llbracket 1, n \rrbracket$. For all $\ell \in \llbracket 0, k-1 \rrbracket$ and $x_k \in \mathbb{R}^{d_{x}}$,

Figures (15)

  • Figure 1: First two dimensions of samples (in red) from each algorithm on the 25 component Gaussian mixture posterior sampling problem with $({d_{x}}, {d_y}) = (100, 1)$. The true posterior samples are given in blue.
  • Figure 2: Sample images for inpainting with center, half, expand masks and for Super Resolution with $4 \times$ and $16 \times$ factors. On the left: FFHQ dataset and on the right ImageNet dataset.
  • Figure 3: Left: JPEG dequantization with QF = $2$. Middle: Poisson denoising. Right: SR $4\times$ Poisson denoising.
  • Figure 4: Trajectory completion where only the middle part of the trajectory is observed. The figures in the $1$st row display $3$ reconstructions per algorithm. The $2$nd and $3$rd rows show confidence intervals across different time steps. The Groundtruth is a trajectory taken from the UCY dataset.
  • Figure 5: Denoising task with Poisson noise on FFHQ.
  • ...and 10 more figures

Theorems & Definitions (3)

  • Proposition 1: informal
  • Proposition 2
  • proof : Proof of \ref{['prop:w2']}