Paul M. Schneider

Damir Filipovic, Paul Schneider

We introduce kernel density machines (KDM), a nonparametric estimator of a Radon--Nikodym derivative, based on reproducing kernel Hilbert spaces. KDM applies to general probability measures on countably generated measurable spaces under minimal assumptions. For computational efficiency, we incorporate a low-rank approximation with precisely controlled error that grants scalability to large-sample settings. We provide rigorous theoretical guarantees, including asymptotic consistency, a functional central limit theorem, and finite-sample error bounds, establishing a strong foundation for practical use. Empirical results based on simulated and real data demonstrate the efficacy and precision of KDM.

Damir Filipovic, Michael Multerer, Paul Schneider

We develop a new framework for estimating joint probability distributions using tensor product reproducing kernel Hilbert spaces (RKHS). Our framework accommodates a low-dimensional, normalized and positive model of a Radon--Nikodym derivative, which we estimate from sample sizes of up to several millions, alleviating the inherent limitations of RKHS modeling. Well-defined normalized and positive conditional distributions are natural by-products to our approach. Our proposal is fast to compute and accommodates learning problems ranging from prediction to classification. Our theoretical findings are supplemented by favorable numerical results.