A non-Markovian approach to diffusion-based sampling

Recently, measure transport via stochastic processes - where samples from a simple prior are evolved toward a target measure specified only by an unnormalized density - has gained significant attention in machine learning and computational statistics. While approaches based on path space measures and time-reversals of diffusions offer rich theoretical insights, their numerical implementation requires simulating entire trajectories, making scaling to high dimensions difficult. Conversely, while recent least-squares "matching" objectives aim to overcome these computational bottlenecks, they introduce significant trade-offs, such as restricting prior distributions or relying on potentially unstable optimization schemes. In this talk, we address these limitations by characterizing these methods as special cases of Markovian and reciprocal projections, developing a novel sampling technique based on suitable fixed-point iterations. Our approach enables learning a stochastic transport map between arbitrary prior and target distributions with a single, scalable, and stable objective. In some sense, it can be interpreted as an extension of the celebrated "stochastic interpolants" to the setting where only an unnormalized density and no empirical data are available. We demonstrate that our method scales to high dimensional problems while preserving mode diversity, achieving state-of-the-art results on complex synthetic distributions and molecular benchmarks. Speaker Bio: Lorenz Richter is a research associate at the Zuse Institute Berlin, where he works on theoretical and computational foundations in machine learning and stochastic analysis. His research spans topics such as diffusion-based generative modeling, stochastic optimal control, neural PDEs, and high-dimensional sampling. He earned a PhD degree in Mathematics from Brandenburgische Technische Universität Cottbus-Senftenberg in 2021. In addition to his academic role, he is the co-founder and CTO of dida, where he leads research and development in applied machine learning. Find seminar details and upcoming talks: https://www.microsoft.com/en-us/research/event/microsoft-research-new-england-generative-modeling-sampling-seminar/