Keep it tighter -- A story on analytical mean embeddings

Linda Chamakh and Zoltan Szabo

LSE Research Online Documents on Economics from London School of Economics and Political Science, LSE Library

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 minimaxoptimal 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-givendefault ratios and on S&P 500 data.

Keywords: kernel methods; divergence measure; optimal transport; portfolio allocation; concentration (search for similar items in EconPapers)
JEL-codes: C1 (search for similar items in EconPapers)
Pages: 47 pages
Date: 2021-10-15
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Persistent link: https://EconPapers.repec.org/RePEc:ehl:lserod:115723

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