In this talk, we study dynamic measure transport for generative modeling, focusing on transport maps that connect a source measure \\(P_0\\) to a target measure \\(P_1\\) by integrating a (conditional) velocity field of the form \\(v_t(x)=\\mathbb{E}[\\dot X_t \\mid X_t = x]\\), where \\(X_\\bullet=(X_t)_t\\) is a stochastic process satisfying \\((X0, X1) \\sim P_0 \\otimes P_1\\) and \\(\\dot X_t\\) is its time derivative. We investigate when \\(X_\\bullet\\) induces a straight-line flow: a flow whose pointwise acceleration vanishes and is therefore exactly integrable by any first-order method. First, we develop multiple characterizations of straight-line flows in terms of PDEs involving the conditional statistics of the process. Then, we prove that straight-line flows under endpoint independence exhibit a sharp dichotomy. On the one hand, we construct explicit, computable straight-line processes for arbitrary Gaussian endpoints. On the other hand, we show that straight-line processes do not exist for targets with sufficiently well-separated modes. We demonstrate this obstruction through a sequence of increasingly general impossibility theorems that uncover a fundamental relationship between the sample-path behavior of a process with independent endpoints and the space-time geometry of this process’ flow map. Taken together, these results provide a structural theory of when straight-line generative flows can, and cannot, exist.<\/p>\n\n\n\n
Panos Tsimpos is a PhD student at the MIT Operations Research Center and the Laboratory for Information and Decision Systems, advised by Youssef Marzouk. He works on probabilistic machine learning, with an emphasis on generative modeling, sampling, and dynamic measure transport. His research focuses on theoretical foundations and algorithmic paradigms, drawing on stochastic analysis, statistics, and mathematical physics. He holds a degree in Mathematics and Physics from Columbia University.<\/p>\n","protected":false},"excerpt":{"rendered":"
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