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Almost Linear Convergence under Minimal Score Assumptions: Quantized Transition Diffusion

Xunpeng Huang, Yingyu Lin, Nikki Lijing Kuang, Hanze Dong, Difan Zou, Yian Ma, Tong Zhang

TL;DR

This work addresses inefficiencies in diffusion-based generative modeling by marrying discrete and continuous diffusion paradigms through Quantized Transition Diffusion (QTD). By histogram-quantizing the target distribution and embedding it into a binary, Hamming-graph space, QTD enables long-range forward transitions with sparse connectivity and employs truncated uniformization for an unbiased, efficient reverse-time sampling. The authors prove a theoretical TV-convergence guarantee with a score-evaluation complexity of $O(d \ln^2(d/\epsilon))$ under minimal score assumptions, advancing the theoretical foundations of diffusion modeling. Overall, QTD offers a principled, scalable framework that unifies discrete and continuous diffusion schemes and yields potential gains in sampling efficiency for high-dimensional data.

Abstract

Continuous diffusion models have demonstrated remarkable performance in data generation across various domains, yet their efficiency remains constrained by two critical limitations: (1) the local adjacency structure of the forward Markov process, which restricts long-range transitions in the data space, and (2) inherent biases introduced during the simulation of time-inhomogeneous reverse denoising processes. To address these challenges, we propose Quantized Transition Diffusion (QTD), a novel approach that integrates data quantization with discrete diffusion dynamics. Our method first transforms the continuous data distribution $p_*$ into a discrete one $q_*$ via histogram approximation and binary encoding, enabling efficient representation in a structured discrete latent space. We then design a continuous-time Markov chain (CTMC) with Hamming distance-based transitions as the forward process, which inherently supports long-range movements in the original data space. For reverse-time sampling, we introduce a \textit{truncated uniformization} technique to simulate the reverse CTMC, which can provably provide unbiased generation from $q_*$ under minimal score assumptions. Through a novel KL dynamic analysis of the reverse CTMC, we prove that QTD can generate samples with $O(d\ln^2(d/ε))$ score evaluations in expectation to approximate the $d$--dimensional target distribution $p_*$ within an $ε$ error tolerance. Our method not only establishes state-of-the-art inference efficiency but also advances the theoretical foundations of diffusion-based generative modeling by unifying discrete and continuous diffusion paradigms.

Almost Linear Convergence under Minimal Score Assumptions: Quantized Transition Diffusion

TL;DR

This work addresses inefficiencies in diffusion-based generative modeling by marrying discrete and continuous diffusion paradigms through Quantized Transition Diffusion (QTD). By histogram-quantizing the target distribution and embedding it into a binary, Hamming-graph space, QTD enables long-range forward transitions with sparse connectivity and employs truncated uniformization for an unbiased, efficient reverse-time sampling. The authors prove a theoretical TV-convergence guarantee with a score-evaluation complexity of under minimal score assumptions, advancing the theoretical foundations of diffusion modeling. Overall, QTD offers a principled, scalable framework that unifies discrete and continuous diffusion schemes and yields potential gains in sampling efficiency for high-dimensional data.

Abstract

Continuous diffusion models have demonstrated remarkable performance in data generation across various domains, yet their efficiency remains constrained by two critical limitations: (1) the local adjacency structure of the forward Markov process, which restricts long-range transitions in the data space, and (2) inherent biases introduced during the simulation of time-inhomogeneous reverse denoising processes. To address these challenges, we propose Quantized Transition Diffusion (QTD), a novel approach that integrates data quantization with discrete diffusion dynamics. Our method first transforms the continuous data distribution into a discrete one via histogram approximation and binary encoding, enabling efficient representation in a structured discrete latent space. We then design a continuous-time Markov chain (CTMC) with Hamming distance-based transitions as the forward process, which inherently supports long-range movements in the original data space. For reverse-time sampling, we introduce a \textit{truncated uniformization} technique to simulate the reverse CTMC, which can provably provide unbiased generation from under minimal score assumptions. Through a novel KL dynamic analysis of the reverse CTMC, we prove that QTD can generate samples with score evaluations in expectation to approximate the --dimensional target distribution within an error tolerance. Our method not only establishes state-of-the-art inference efficiency but also advances the theoretical foundations of diffusion-based generative modeling by unifying discrete and continuous diffusion paradigms.

Paper Structure

This paper contains 24 sections, 17 theorems, 193 equations, 3 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quantized Transition Diffusion
  4. Histogram Approximation
  5. Binary Encoding of the Discrete Space
  6. The forward CTMC starting from $q_*$.
  7. Truncated Uniformization
  8. Theoretical Results
  9. Conclusion and Limitation
  10. Notation Summary
  11. Technical Lemmas
  12. Forward and Reverse Processes of Discrete Diffusion Models
  13. The Forward Process of Discrete Diffusion Models
  14. Matrix Presentation.
  15. Proof of Eq. \ref{['eq:rev_func']}
  16. ...and 9 more sections

Key Result

Lemma 3.1

Suppose the target distribution $p_*\propto \exp(-f_*)$ is $\sigma$ sub-Gaussian and $f_*$ is $H$--smooth, we can construct $\overline{p}_*$ defined on a finite cube $\mathrm{Cube}\left(L\right)$ with length to satisfy $\mathrm{TV}\left(p_*, \overline{p}_*\right)\le 3\epsilon$.

Figures (3)

  • Figure 1: Visualization of different adjacency structures. The bold blue edges highlight a diameter path---a shortest path between the two most distant vertices in each graph. Drawing samples in a discrete space ${\mathcal{Y}}$ by simulating a CTMC can be viewed as traversing a graph whose diameter governs the number of iterations required for convergence, while the out-degree of each node influences the per-iteration complexity. In the neighborhood adjacency $G_{\text{Tridiagonal}}$, each node has an out-degree of $O(1)$ but a diameter of $O(|{\mathcal{Y}}|)$. For the dense adjacency, the graph $G_{\text{Dense}}$ attains a diameter of $O(1)$ at the cost of an $O(|{\mathcal{Y}}|)$ out-degree. Notably, the binary adjacency $G_{\text{Hypercube}}$ offers a balanced design, featuring both a diameter and an out-degree of $O(\log|{\mathcal{Y}}|)$.
  • Figure 2: Visualization of the histogram approximation. The first step regularizes the original distribution in some bounded sets but controls the TV gap by Lemma \ref{['lem:histog_dis_comp']}. The second step quantizes the probability density to a histogram-like distribution but controls the TV gap by Lemma \ref{['lem:histog_dis_quan']}.
  • Figure 3: Heatmaps of the probability transition at different time steps $t$ for four diffusion processes: a continuous normalized Gaussian kernel on $[0,1]$ (top row), and discrete CTMCs over $|{\mathcal{Y}}| = 8$ states based on tridiagonal, dense, and hypercube transition rate matrices (bottom three rows).

Theorems & Definitions (33)

  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Theorem 4.1
  • Lemma B.1: Theorem 4.10 of boucheron2003concentration
  • Lemma B.2: Chain rule of TV
  • Lemma B.3: Backward Kolmogorov equation
  • Lemma B.4: Lemma 11 in vempala2019rapid
  • Lemma C.1: Forward transition kernel
  • Remark C.2
  • ...and 23 more