On the Contractivity of Stochastic Interpolation Flow

2025-04-14

AI Summary

This work addresses the Lipschitz continuity of stochastic interpolation flows mapping a Gaussian source distribution to a strongly log-concave target distribution. Methodologically, it integrates stochastic differential equations, optimal transport theory, and log-concave analysis to rigorously characterize the regularity of such flows. The key contribution is the first proof that this stochastic interpolation map attains the same sharp Lipschitz constant as Caffarelli’s deterministic optimal transport map—thereby extending the Lipschitz transport framework beyond Gaussian sources to general strongly log-concave targets. The established strict contraction property provides tight theoretical guarantees for stability, estimation error control, and generative reliability in high-dimensional stochastic interpolation sampling. These results significantly enhance sampling efficiency and generalization capacity, offering foundational insights for scalable and robust sampling algorithms in non-Gaussian settings.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Learning with Manifolds
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

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.

Problem

• Analyzing Lipschitz properties of stochastic interpolation flows
• Constructing transport maps between non-Gaussian distributions
• Improving sampling and estimation in high-dimensional spaces

Innovation

• Stochastic interpolation for high-dimensional sampling
• Lipschitz flow map with Gaussian base
• Constructs Lipschitz transport maps non-Gaussian distributions