Keep it Tighter -- A Story on Analytical Mean Embeddings

TL;DR

This paper introduces a new estimator for maximum mean discrepancy (MMD) that leverages analytical mean embeddings, providing tighter concentration bounds and improved efficiency for one-dimensional distributions, with applications in finance.

Contribution

It develops a semi-explicit MMD estimator for distributions on the real line, achieving tighter concentration bounds and extending to unbounded kernels with minimax-optimal bounds.

Findings

Tighter concentration bounds for the proposed estimator.

Extension to unbounded exponential kernels with optimal bounds.

Successful application to real-world financial data.

Abstract

Kernel techniques are among the most popular and flexible approaches in data science allowing to represent probability measures without loss of information under mild conditions. The resulting mapping called mean embedding gives rise to a divergence measure referred to as maximum mean discrepancy (MMD) with existing quadratic-time estimators (w.r.t. the sample size) and known convergence properties for bounded kernels. In this paper we focus on the problem of MMD estimation when the mean embedding of one of the underlying distributions is available analytically. Particularly, we consider distributions on the real line (motivated by financial applications) and prove tighter concentration for the proposed estimator under this semi-explicit setting; we also extend the result to the case of unbounded (exponential) kernel with minimax-optimal lower bounds. We demonstrate the efficiency of our approach beyond synthetic example in three real-world examples relying on one-dimensional random variables: index replication and calibration on loss-given-default ratios and on S&P 500 data.