Almost Linear Convergence under Minimal Score Assumptions: Quantized Transition Diffusion

• URL: http://arxiv.org/abs/2505.21892v1
• Date: Wed, 28 May 2025 02:10:11 GMT
• Title: Almost Linear Convergence under Minimal Score Assumptions: Quantized Transition Diffusion
• Authors: Xunpeng Huang, Yingyu Lin, Nikki Lijing Kuang, Hanze Dong, Difan Zou, Yian Ma, Tong Zhang,
• Abstract summary: We propose Quantized Transition Diffusion (QTD), a novel approach that integrates data quantization with discrete diffusion dynamics.<n>Our method first transforms the continuous data distribution $p_*$ into a discrete one $q_*$ via histogram approximation and binary encoding.<n>For reverse-time sampling, we introduce a textittruncated uniformization technique to simulate the reverse CTMC.
• Score: 25.542593757387095
• License: http://creativecommons.org/licenses/by/4.0/
• 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 $O(d\ln^2(d/\epsilon))$ score evaluations in expectation to approximate the $d$--dimensional target distribution $p_*$ within an $\epsilon$ 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.

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