Submitted Paper
Inserted: 24 jun 2025
Last Updated: 24 jun 2025
Year: 2025
Abstract:
The mathematical theory of a novel variational approximation scheme for general second and fourth order diffusion equations (GF) is developed; the Kernel-Density-Estimator Minimizing Movement Scheme (KDE-MM-Scheme) preserves the structure of (GF) as a steepest descent with regard to an energy functional and a Wasserstein distance in the space of probability measures, at the same time imitating the corresponding motion of a finite number of particles (data points) on a discrete timescale. Roughly speaking, the KDE-MM-Scheme constitutes a simplification of the classical Minimizing Movement scheme for (GF) (often referred to as `JKO scheme'), in which the corresponding minimum problems are relaxed and restricted to the values of Kernel Density Estimators each associated with a finite dataset. Rigorous mathematical proofs show that the KDE-MM-Scheme yields solutions to (GF) if we let the time step sizes and the dataset sizes (particle numbers) simultaneously go to zero and infinity respectively. Uniting abstract analysis in metric spaces with application-orientated concepts from statistics and machine learning, our examinations will form the mathematical foundation for a novel computationally tractable algorithm approximating solutions to (GF). A particular ingredient for our analysis is a general and robust stability theory for discrete-time steepest descents under the occurrence of Gamma-perturbations of the energy functional in the Minimizing Movement scheme and relaxations of the corresponding minimum problems.
This paper is a shortened form of the author's arXiv preprint from October 2023.
Keywords: Optimal transport, $\Gamma$-convergence, Gradient flows, minimizing movements, Machine learning, diffusion equations, kernel density estimation
Download: