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Linear Convergence of Diffusion Models Under the Manifold Hypothesis

Peter Potaptchik, Iskander Azangulov, George Deligiannidis

TL;DR

This work analyzes score-based diffusion models under the manifold hypothesis, where data lie on a d-dimensional manifold within a high-dimensional space. By combining a martingale-based view of the backward process with a discretization tailored to the intrinsic dimension, the authors prove that the number of denoising steps needed to achieve KL convergence scales linearly with d up to logarithmic factors. They also establish the tightness of this linear dependence via a tensorization argument, showing that the bound cannot in general be improved with respect to d. The results help explain why diffusion models perform well in practice on data with low intrinsic dimensionality, despite large ambient dimensions.

Abstract

Score-matching generative models have proven successful at sampling from complex high-dimensional data distributions. In many applications, this distribution is believed to concentrate on a much lower $d$-dimensional manifold embedded into $D$-dimensional space; this is known as the manifold hypothesis. The current best-known convergence guarantees are either linear in $D$ or polynomial (superlinear) in $d$. The latter exploits a novel integration scheme for the backward SDE. We take the best of both worlds and show that the number of steps diffusion models require in order to converge in Kullback-Leibler~(KL) divergence is linear (up to logarithmic terms) in the intrinsic dimension $d$. Moreover, we show that this linear dependency is sharp.

Linear Convergence of Diffusion Models Under the Manifold Hypothesis

TL;DR

This work analyzes score-based diffusion models under the manifold hypothesis, where data lie on a d-dimensional manifold within a high-dimensional space. By combining a martingale-based view of the backward process with a discretization tailored to the intrinsic dimension, the authors prove that the number of denoising steps needed to achieve KL convergence scales linearly with d up to logarithmic factors. They also establish the tightness of this linear dependence via a tensorization argument, showing that the bound cannot in general be improved with respect to d. The results help explain why diffusion models perform well in practice on data with low intrinsic dimensionality, despite large ambient dimensions.

Abstract

Score-matching generative models have proven successful at sampling from complex high-dimensional data distributions. In many applications, this distribution is believed to concentrate on a much lower -dimensional manifold embedded into -dimensional space; this is known as the manifold hypothesis. The current best-known convergence guarantees are either linear in or polynomial (superlinear) in . The latter exploits a novel integration scheme for the backward SDE. We take the best of both worlds and show that the number of steps diffusion models require in order to converge in Kullback-Leibler~(KL) divergence is linear (up to logarithmic terms) in the intrinsic dimension . Moreover, we show that this linear dependency is sharp.

Paper Structure

This paper contains 15 sections, 16 theorems, 69 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Concurrent work:
  3. Preliminaries
  4. Diffusion Models
  5. Assumptions and Notation
  6. Discretization Scheme
  7. Main Results
  8. Tightness of linear bound.
  9. Proof of Theorem \ref{['thm:main']}
  10. Conclusion & Future Work
  11. Elements of Manifold Learning
  12. Discretization Coefficients
  13. Failure of Exponential Integrator
  14. Modified Discretization Scheme
  15. Proofs of Lemmas

Key Result

theorem 1

Let $\mu$ be a measure satisfying Assumptions asmp:1--asmp:3. Let $t > 0$ and let $X_t = c_tX_0 +\sigma_t Z_D$, where $X_0\sim \mu$ and $Z_D\sim \mathcal{N}\left(0, \mathop{\mathrm{\mathrm{Id}}}\nolimits_D\right)$ are independent. Then with probability at least $1-\delta$

Figures (2)

  • Figure 1: Illustration of a typical error of classic Discretization scheme. While point $c^{-1}_tX_t + c^{-1}_t\sigma_t s_t(X_t)$ is $O\left(\sqrt{d}\sigma_t/c_t\right)$ close to $X_0$, the difference $X_{t+\gamma}-c_\gamma X_t\sim \mathcal{N}\left(0,\sigma_\gamma\mathop{\mathrm{\mathrm{Id}}}\nolimits_D\right)$, which is of order $O\left(\sigma_\gamma\sqrt{D}\right)$. So, for $\gamma/t \lesssim d/D$ the error scales as at least $\sigma_\gamma\sqrt{D}$.
  • Figure :

Theorems & Definitions (26)

  • remark 1
  • remark 2
  • theorem 1
  • proposition 1
  • definition 1
  • theorem 2
  • corollary 1
  • lemma 1
  • lemma 2
  • lemma 3
  • ...and 16 more