Learning L\'evy density via adaptive RKHS regression with bi-level optimization

TL;DR

This paper introduces an adaptive RKHS regression method with bilevel optimization to accurately learn Le9vy densities from probability density data, improving robustness and efficiency over traditional regularization techniques.

Contribution

It develops a novel adaptive RKHS framework combined with a GSVD-based bilevel optimization algorithm for regularization parameter selection in Le9vy density estimation.

Findings

The method achieves near optimal error decay rates.

It outperforms classical L-curve and GCV strategies.

The adaptive RKHS norm yields more accurate and robust estimates.

Abstract

We propose a nonparametric method to learn the L\'evy density from probability density data governed by a nonlocal Fokker-Planck equation. We recast the problem as identifying the kernel in a nonlocal integral operator from discrete data, which leads to an ill-posed inverse problem. To regularize it, we construct an adaptive reproducing kernel Hilbert space (RKHS) whose kernel is built directly from the data. Under standard source and spectral decay conditions, we show that the reconstruction error decays in the mesh size at a near optimal rate. Importantly, we develop a generalized singular value decomposition (GSVD)-based bilevel optimization algorithm to choose the regularization parameter, leading to efficient and robust computation of the regularized estimator. Numerical experiments for several L\'evy densities, drift fields and data types (PDE-based densities and sample ensemble-based KDE reconstructions) demonstrate that our bilevel RKHS method outperforms classical L-curve and generalized cross-validation strategies and that the adaptive RKHS norm is more accurate and robust than $L^2_\rho$- and $\ell^2$-based regularization.