  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.
