Discrete generative diffusion models without stochastic differential equations: A tensor network approach.

Area of Science:

• Machine Learning
• Statistical Physics
• Computational Science

Background:

• Diffusion models (DMs) generate data by reversing noise addition via learned score functions.
• Standard DMs operate on continuous distributions using stochastic differential equations.
• Lattice systems with discrete degrees of freedom pose challenges for standard DM approaches.

Purpose of the Study:

• Generalize diffusion models to discrete lattice systems.
• Develop an efficient sampling method for discrete data using tensor networks.
• Integrate discrete diffusion models with Monte Carlo methods for statistical physics applications.

Main Methods:

• Parametrized data and evolution operators as tensor networks (TNs).
• Developed discrete diffusion models (DDMs) leveraging Markov chain jump dynamics.
• Utilized the auto-regressive properties of TNs for sample generation.

Main Results:

• Exact representation of denoising dynamics using TNs.
• Efficient and unbiased sample generation from discrete distributions.
• Constructed an efficient learning scheme for Boltzmann-like distributions via TNs and Monte Carlo integration.

Conclusions:

• Tensor networks provide an efficient framework for discrete diffusion models.
• The proposed method enables accurate sampling of complex discrete distributions.
• Demonstrated applicability to studying equilibrium in models with non-trivial thermodynamics.