Bit-Level Discrete Diffusion with Markov Probabilistic Models: An Improved Framework with Sharp Convergence Bounds under Minimal Assumptions

TL;DR

This paper presents a new discrete diffusion algorithm using Markov chains for data generation, with proven convergence bounds and competitive performance on binary datasets, bridging theory and practice.

Contribution

Introduces Discrete Markov Probabilistic Models with theoretical convergence guarantees and practical effectiveness for discrete data generation.

Findings

Proven convergence bounds under minimal assumptions.

Effective generation of binary and Bernoulli data.

Competitive results on high-dimensional binary datasets.

Abstract

This paper introduces Discrete Markov Probabilistic Models (DMPMs), a novel discrete diffusion algorithm for discrete data generation. The algorithm operates in discrete bit space, where the noising process is a continuous-time Markov chain that flips labels uniformly at random. The time-reversal process, like the forward noise process, is a jump process with its intensity governed by a discrete analogue of the classical score function. Crucially, this intensity is proven to be the conditional expectation of a function of the forward process, underlining theoretical alignment with score-based generative models. We establish convergence bounds for the algorithm under minimal assumptions, ensuring robustness and efficiency, which we demonstrate through experiments on low-dimensional Bernoulli-distributed datasets and high-dimensional binary MNIST data. The results highlight competitive performance in generating discrete structures compared to the state-of-the-art. This work bridges theoretical foundations and practical applications, advancing the development of effective and theoretically grounded discrete generative modeling.