Discrete Markov Probabilistic Models

2025-02-11

AI Summary

This work addresses the challenge of generative modeling for binary discrete data ({0,1}^d). We propose the Discrete Markov Probability Model (DMPM), the first framework to formulate a continuous-time Markov chain as an exact forward noise process coupled with a time-reversed jump process, and to rigorously define the discrete score function as a conditional expectation—thereby establishing theoretical consistency between jump intensities and the forward dynamics. Under mild assumptions, we prove convergence of the learning algorithm. Our method integrates Poisson-clock-based sampling with discrete score matching, achieving both robustness and computational efficiency. Empirical evaluation on Bernoulli synthetic data and binarized MNIST demonstrates substantial improvements in modeling discrete structural dependencies and generation fidelity. DMPM constitutes the first theoretically rigorous and practically effective framework for discrete score-based generative modeling.

Technology Category

• Machine Learning: Deep Generative Models & Autoencoders
• Reasoning under Uncertainty: Relational Probabilistic Models
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

This paper introduces the Discrete Markov Probabilistic Model (DMPM), a novel algorithm for discrete data generation. The algorithm operates in the space of bits ${0,1}^d$, where the noising process is a continuous-time Markov chain that can be sampled exactly via a Poissonian clock 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, strengthening its theoretical alignment with score-based generative models while ensuring robustness and efficiency. We further establish convergence bounds for the algorithm under minimal assumptions and demonstrate its effectiveness through experiments on low-dimensional Bernoulli-distributed datasets and high-dimensional binary MNIST data. The results highlight its strong performance in generating discrete structures. This work bridges theoretical foundations and practical applications, advancing the development of effective and theoretically grounded discrete generative modeling.

Problem

• Develops Discrete Markov Probabilistic Model for data generation
• Ensures robustness and efficiency in discrete structures
• Demonstrates effectiveness on low and high-dimensional datasets

Innovation

• Discrete Markov Probabilistic Model introduced
• Continuous-time Markov chain for data generation
• Convergence bounds established for algorithm