On the Contractivity of Stochastic Interpolation Flow

• URL: http://arxiv.org/abs/2504.10653v1
• Date: Mon, 14 Apr 2025 19:10:22 GMT
• Title: On the Contractivity of Stochastic Interpolation Flow
• Authors: Max Daniels,
• Abstract summary: We investigate, a recently introduced framework for high dimensional sampling which bears many similarities to diffusion modeling.<n>We show that for a base distribution and a strongly log-concave target distribution, the flow map is Lipschitz with a sharp constant which matches that of Caffarelli's theorem for optimal transport maps.<n>We are further able to construct Lipschitz transport maps between non-Gaussian distributions, generalizing some recent constructions in the literature on transport methods for establishing functional inequalities.
• Score: 1.90365714903665
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• Abstract: We investigate stochastic interpolation, a recently introduced framework for high dimensional sampling which bears many similarities to diffusion modeling. Stochastic interpolation generates a data sample by first randomly initializing a particle drawn from a simple base distribution, then simulating deterministic or stochastic dynamics such that in finite time the particle's distribution converges to the target. We show that for a Gaussian base distribution and a strongly log-concave target distribution, the stochastic interpolation flow map is Lipschitz with a sharp constant which matches that of Caffarelli's theorem for optimal transport maps. We are further able to construct Lipschitz transport maps between non-Gaussian distributions, generalizing some recent constructions in the literature on transport methods for establishing functional inequalities. We discuss the practical implications of our theorem for the sampling and estimation problems required by stochastic interpolation.

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