Score-based constrained generative modeling via Langevin diffusions with boundary conditions

TL;DR

This paper introduces a new constrained generative modeling approach using Langevin dynamics with boundary reflections, improving sampling accuracy under constraints and providing efficient numerical algorithms.

Contribution

It proposes a novel constrained generative model with specular reflection in Langevin dynamics and compares it with existing reflected SDE models, offering efficient sampling methods.

Findings

Efficient numerical samplers with optimal convergence rates.

Enhanced constrained sampling using specular reflection.

Comprehensive comparison of Langevin and reflected SDE models.

Abstract

Score-based generative models based on stochastic differential equations (SDEs) achieve impressive performance in sampling from unknown distributions, but often fail to satisfy underlying constraints. We propose a constrained generative model using kinetic (underdamped) Langevin dynamics with specular reflection of velocity on the boundary defining constraints. This results in piecewise continuously differentiable noising and denoising process where the latter is characterized by a time-reversed dynamics restricted to a domain with boundary due to specular boundary condition. In addition, we also contribute to existing reflected SDEs based constrained generative models, where the stochastic dynamics is restricted through an abstract local time term. By presenting efficient numerical samplers which converge with optimal rate in terms of discretizations step, we provide a comprehensive comparison of models based on confined (specularly reflected kinetic) Langevin diffusion with models based on reflected diffusion with local time.