Random-Bridges as Stochastic Transports for Generative Models

Abstract

This paper motivates the use of random-bridges -- stochastic processes conditioned to take target distributions at fixed timepoints -- in the realm of generative modelling. Herein, random-bridges act as stochastic transports between two probability distributions when appropriately initialized, displaying either Markovian or non-Markovian, and continuous, discontinuous, or hybrid patterns depending on the choice of the driving process. We show how one can start from general probabilistic statements and branch out into specific representations for learning and simulation algorithms. Essentially, our framework formulates stochastic transports as noisy information processes governing the dynamic translation and propagation of information between arbitrary probability measures. Our theoretical and algorithmic presentation details training and simulation frameworks for both Gaussian and Lévy (specifically, Gamma, Poisson and Stable-$1/2$) random-bridge transports. The empirical validation focuses on the Gaussian case for demonstration and comparison purposes, where our construct delivers high-quality samples, achieving competitive Fréchet Inception Distance scores compared to widely used denoising diffusions and flow matching frameworks.