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Adapting to Unknown Low-Dimensional Structures in Score-Based Diffusion Models

Gen Li, Yuling Yan

TL;DR

This work analyzes score-based diffusion models when the data distribution concentrates on or near a low-dimensional manifold. By designing a specific coefficient schedule for the DDPM sampler and developing a deterministic analysis that decouples discretization from score estimation error, the authors prove a dimension-free convergence rate (up to log factors) that depends on the intrinsic dimension k rather than the ambient dimension d. They also show a near-unique coefficient design is necessary to avoid ambient-dimension discretization errors, and provide a detailed five-step proof strategy leveraging high-probability sets and KL bounds. Simulation results corroborate the theory, illustrating dimension-independent behavior under the proposed design even as ambient dimension grows. The work thus establishes the first theoretical guarantee that DDPM samplers can adapt to unknown low-dimensional data structures, with implications for understanding and improving generative diffusion models in high-dimensional settings.

Abstract

This paper investigates score-based diffusion models when the underlying target distribution is concentrated on or near low-dimensional manifolds within the higher-dimensional space in which they formally reside, a common characteristic of natural image distributions. Despite previous efforts to understand the data generation process of diffusion models, existing theoretical support remains highly suboptimal in the presence of low-dimensional structure, which we strengthen in this paper. For the popular Denoising Diffusion Probabilistic Model (DDPM), we find that the dependency of the error incurred within each denoising step on the ambient dimension $d$ is in general unavoidable. We further identify a unique design of coefficients that yields a converges rate at the order of $O(k^{2}/\sqrt{T})$ (up to log factors), where $k$ is the intrinsic dimension of the target distribution and $T$ is the number of steps. This represents the first theoretical demonstration that the DDPM sampler can adapt to unknown low-dimensional structures in the target distribution, highlighting the critical importance of coefficient design. All of this is achieved by a novel set of analysis tools that characterize the algorithmic dynamics in a more deterministic manner.

Adapting to Unknown Low-Dimensional Structures in Score-Based Diffusion Models

TL;DR

This work analyzes score-based diffusion models when the data distribution concentrates on or near a low-dimensional manifold. By designing a specific coefficient schedule for the DDPM sampler and developing a deterministic analysis that decouples discretization from score estimation error, the authors prove a dimension-free convergence rate (up to log factors) that depends on the intrinsic dimension k rather than the ambient dimension d. They also show a near-unique coefficient design is necessary to avoid ambient-dimension discretization errors, and provide a detailed five-step proof strategy leveraging high-probability sets and KL bounds. Simulation results corroborate the theory, illustrating dimension-independent behavior under the proposed design even as ambient dimension grows. The work thus establishes the first theoretical guarantee that DDPM samplers can adapt to unknown low-dimensional data structures, with implications for understanding and improving generative diffusion models in high-dimensional settings.

Abstract

This paper investigates score-based diffusion models when the underlying target distribution is concentrated on or near low-dimensional manifolds within the higher-dimensional space in which they formally reside, a common characteristic of natural image distributions. Despite previous efforts to understand the data generation process of diffusion models, existing theoretical support remains highly suboptimal in the presence of low-dimensional structure, which we strengthen in this paper. For the popular Denoising Diffusion Probabilistic Model (DDPM), we find that the dependency of the error incurred within each denoising step on the ambient dimension is in general unavoidable. We further identify a unique design of coefficients that yields a converges rate at the order of (up to log factors), where is the intrinsic dimension of the target distribution and is the number of steps. This represents the first theoretical demonstration that the DDPM sampler can adapt to unknown low-dimensional structures in the target distribution, highlighting the critical importance of coefficient design. All of this is achieved by a novel set of analysis tools that characterize the algorithmic dynamics in a more deterministic manner.
Paper Structure (40 sections, 12 theorems, 156 equations, 1 figure)

This paper contains 40 sections, 12 theorems, 156 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Diffusion models
  3. Inadequacy of existing results
  4. Our contributions
  5. Problem set-up
  6. Forward process.
  7. Score functions.
  8. The DDPM sampler.
  9. Target data distribution.
  10. Learning rate schedule.
  11. Main results
  12. Convergence analysis
  13. Uniqueness of coefficient design
  14. Analysis for the DDPM sampler (Proof of Theorem \ref{['thm:SDE']})
  15. Step 1: identifying high-probability sets
  16. ...and 25 more sections

Key Result

Theorem 1

Suppose that we take the coefficients for the DDPM sampler (eq:DDPM) to be $\eta_{t}=\eta_{t}^{\star}$ and $\sigma_{t}=\sigma_{t}^{\star}$ (cf. (eq:defn-step-size)), then there exists some universal constant $C>0$ such that

Figures (1)

  • Figure 1: The KL divergence between $q_1$ and $p_1$ for $T \in \{100,200,500,1000\}$, when $p_\mathsf{data}=\mathcal{N}(0,I_k)$. We fix the low intrinsic dimension $k=8$, and let the ambient dimension $d$ grow from $10$ to $1000$.

Theorems & Definitions (29)

  • Theorem 1
  • Theorem 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • ...and 19 more