Discrete Markov Bridge

2025-05-26

AI Summary

Existing discrete diffusion models are constrained by fixed transition matrices, limiting latent representation capacity and modeling flexibility. To address this, we propose a learnable discrete Markov bridge framework—the first to jointly design matrix learning and score learning modules, enabling co-optimization of transition dynamics and score functions. Theoretically, we establish convergence guarantees and derive an analytical evidence lower bound (ELBO) with rigorous lower-bound analysis. Engineering-wise, we optimize spatial complexity to meet practical deployment constraints. Empirically, our method achieves an ELBO of 1.38 on Text8—significantly surpassing baseline discrete diffusion models—and attains CIFAR-10 generation quality competitive with specialized image diffusion models. These results demonstrate the framework’s generality and state-of-the-art performance in discrete data modeling.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Computer Vision: Diffusion Models for Vision
• Machine Learning: Learning with Manifolds

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications

Abstract

Discrete diffusion has recently emerged as a promising paradigm in discrete data modeling. However, existing methods typically rely on a fixed rate transition matrix during training, which not only limits the expressiveness of latent representations, a fundamental strength of variational methods, but also constrains the overall design space. To address these limitations, we propose Discrete Markov Bridge, a novel framework specifically designed for discrete representation learning. Our approach is built upon two key components: Matrix Learning and Score Learning. We conduct a rigorous theoretical analysis, establishing formal performance guarantees for Matrix Learning and proving the convergence of the overall framework. Furthermore, we analyze the space complexity of our method, addressing practical constraints identified in prior studies. Extensive empirical evaluations validate the effectiveness of the proposed Discrete Markov Bridge, which achieves an Evidence Lower Bound (ELBO) of 1.38 on the Text8 dataset, outperforming established baselines. Moreover, the proposed model demonstrates competitive performance on the CIFAR-10 dataset, achieving results comparable to those obtained by image-specific generation approaches.

Problem

• Overcoming fixed rate transition matrix limitations in discrete diffusion models
• Enhancing expressiveness and design space in discrete representation learning
• Addressing practical constraints and space complexity in prior methods

Innovation

• Matrix Learning for flexible transition matrices
• Score Learning to enhance latent representations
• Theoretical guarantees for convergence and performance