Score-based constrained generative modeling via Langevin diffusions with boundary conditions
This addresses the challenge of generating constrained samples in machine learning, though it appears incremental as it builds on existing reflected SDE-based models.
The paper tackles the problem of score-based generative models failing to satisfy constraints by proposing a constrained generative model using kinetic Langevin dynamics with specular reflection on boundaries, resulting in efficient numerical samplers with optimal convergence rates.
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.