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Restoration-Degradation Beyond Linear Diffusions: A Non-Asymptotic Analysis For DDIM-Type Samplers

Sitan Chen, Giannis Daras, Alexandros G. Dimakis

TL;DR

This work develops a non-asymptotic theory for deterministic diffusion samplers by reframing the probability flow ODE as a two-step process: a restoration step that refines past-state information via gradient ascent on the conditional likelihood, and a degradation step that advances along the forward dynamics with noise aligned to the current iterate. It generalizes DDIM to non-linear forward processes and proves polynomial convergence guarantees under mild data-distribution assumptions, yielding a first non-asymptotic analysis for deterministic samplers. The results rely on an interpolation-based analytic framework that bounds the KL divergence between the discretized sampler and the true reverse process, and extend to Euler discretizations as well. Together, these insights provide a rigorous, scalable pathway to understanding and improving fast, deterministic diffusion samplers with broad applicability across non-linear forward models.

Abstract

We develop a framework for non-asymptotic analysis of deterministic samplers used for diffusion generative modeling. Several recent works have analyzed stochastic samplers using tools like Girsanov's theorem and a chain rule variant of the interpolation argument. Unfortunately, these techniques give vacuous bounds when applied to deterministic samplers. We give a new operational interpretation for deterministic sampling by showing that one step along the probability flow ODE can be expressed as two steps: 1) a restoration step that runs gradient 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 iterate. This perspective allows us to extend denoising diffusion implicit models to general, non-linear forward processes. We then develop the first polynomial convergence bounds for these samplers under mild conditions on the data distribution.

Restoration-Degradation Beyond Linear Diffusions: A Non-Asymptotic Analysis For DDIM-Type Samplers

TL;DR

This work develops a non-asymptotic theory for deterministic diffusion samplers by reframing the probability flow ODE as a two-step process: a restoration step that refines past-state information via gradient ascent on the conditional likelihood, and a degradation step that advances along the forward dynamics with noise aligned to the current iterate. It generalizes DDIM to non-linear forward processes and proves polynomial convergence guarantees under mild data-distribution assumptions, yielding a first non-asymptotic analysis for deterministic samplers. The results rely on an interpolation-based analytic framework that bounds the KL divergence between the discretized sampler and the true reverse process, and extend to Euler discretizations as well. Together, these insights provide a rigorous, scalable pathway to understanding and improving fast, deterministic diffusion samplers with broad applicability across non-linear forward models.

Abstract

We develop a framework for non-asymptotic analysis of deterministic samplers used for diffusion generative modeling. Several recent works have analyzed stochastic samplers using tools like Girsanov's theorem and a chain rule variant of the interpolation argument. Unfortunately, these techniques give vacuous bounds when applied to deterministic samplers. We give a new operational interpretation for deterministic sampling by showing that one step along the probability flow ODE can be expressed as two steps: 1) a restoration step that runs gradient 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 iterate. This perspective allows us to extend denoising diffusion implicit models to general, non-linear forward processes. We then develop the first polynomial convergence bounds for these samplers under mild conditions on the data distribution.
Paper Structure (26 sections, 25 theorems, 131 equations)

This paper contains 26 sections, 25 theorems, 131 equations.

Table of Contents

  1. Introduction
  2. Roadmap.
  3. Preliminaries
  4. Operational Interpretation for the Probability Flow ODE
  5. Warmup: linear SDEs and DDIM
  6. General diffusions
  7. Restoration operator.
  8. Restoration operator as gradient ascent.
  9. Degradation operator.
  10. Extensions to other samplers
  11. Discretization Analysis
  12. DDIM-type sampler
  13. Statement of results
  14. Proof overview
  15. Related Work
  16. ...and 11 more sections

Key Result

Theorem 1.1

Denote by $h$ the infinitesimally small step size with which we discretize the probability flow ODE. Let $\ell\in\mathbb{N}$ be a parameter for which $\ell \to \infty$ and $\ell h \to 0$. For any forward process, running the probability flow ODE for time $h$ is equivalent to running the following tw

Theorems & Definitions (48)

  • Theorem 1.1: Informal, see Section \ref{['sec:interpret']}
  • Theorem 1.2: Informal, see Theorem \ref{['thm:main']}
  • Remark 1
  • Theorem 4.1
  • Remark 2
  • Corollary 4.2
  • Theorem 4.3
  • Lemma 4.4: See Lemmas \ref{['lem:laplace']} and \ref{['lem:laplace2']}
  • proof : Proof sketch
  • Theorem C.1
  • ...and 38 more