Continuous diffusion models are commonly acknowledged to display a deterministic probability flow, whereas discrete diffusion models do not. In this paper, we aim to establish the fundamental theory for the probability flow of discrete diffusion models. Specifically, we first prove that the continuous probability flow is the Monge optimal transport map under certain conditions, and also present an equivalent evidence for discrete cases. In view of these findings, we are then able to define the discrete probability flow in line with the principles of optimal transport. Finally, drawing upon our newly established definitions, we propose a novel sampling method that surpasses previous discrete diffusion models in its ability to generate more certain outcomes. Extensive experiments on the synthetic toy dataset and the CIFAR-10 dataset have validated the effectiveness of our proposed discrete probability flow. Code is released at: https://github.com/PangzeCheung/Discrete-Probability-Flow.

翻译：连续扩散模型通常被认为具有确定性概率流，而离散扩散模型则不具备这一特性。本文旨在建立离散扩散模型概率流的基础理论。具体而言，我们首先证明在特定条件下连续概率流即为Monge最优输运映射，同时为离散情形提供了等价性证明。基于这些发现，我们得以依据最优输运原理定义离散概率流。最后，基于新建立的定义，我们提出了一种新型采样方法，该方法在生成更确定结果方面优于先前的离散扩散模型。在合成玩具数据集和CIFAR-10数据集上的大量实验验证了所提出的离散概率流的有效性。代码发布在：https://github.com/PangzeCheung/Discrete-Probability-Flow。