Submitted Paper

Inserted: 19 oct 2023
Last Updated: 19 oct 2023

Year: 2023

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 gradient flow with regard to an energy functional and a Wasserstein distance in the space of probability measures, at the same time imitating the steepest descent 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 (weak) 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 theory is a thorough and general analysis of the JKO scheme under the occurrence of $\Gamma$-perturbations $\phi_n$ of the energy functional $\phi$. The discrete-time steepest descents w.r.t. $\phi_n, \ n\in\mathbb{N},$ are directly linked with (GF) through an appropriate correlation between time step sizes and parameters that only depends on the velocity of $\Gamma$-convergence $\phi_n\stackrel{\Gamma}{\to}\phi$.

Keywords: Optimal transport, $\Gamma$-convergence, Gradient flows, minimizing movements, Machine learning, diffusion equations, kernel density estimation