On the Contractivity of Stochastic Interpolation Flow
This work provides theoretical guarantees for sampling methods in machine learning, but it is incremental as it builds on existing stochastic interpolation and transport map frameworks.
The paper tackles the problem of high-dimensional sampling by analyzing the contractivity of stochastic interpolation flows, showing that for Gaussian base and strongly log-concave target distributions, the flow map is Lipschitz with a sharp constant matching Caffarelli's theorem, and extends this to construct Lipschitz transport maps for non-Gaussian distributions.
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.