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A rigorous theory of conditional mean embeddings

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Klebanov, Ilja, Schuster, Ingmar and Sullivan, T. J. (2020) A rigorous theory of conditional mean embeddings. SIAM Journal on Mathematics of Data Science, 2 (3). pp. 583-606. doi:10.1137/19M1305069

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Official URL: http://dx.doi.org/10.1137/19M1305069

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Abstract

Conditional mean embeddings (CMEs) have proven themselves to be a powerful tool in many machine learning applications. They allow the efficient conditioning of probability distributions within the corresponding reproducing kernel Hilbert spaces by providing a linear-algebraic relation for the kernel mean embeddings of the respective joint and conditional probability distributions. Both centered and uncentered covariance operators have been used to define CMEs in the existing literature. In this paper, we develop a mathematically rigorous theory for both variants, discuss the merits and problems of each, and significantly weaken the conditions for applicability of CMEs. In the course of this, we demonstrate a beautiful connection to Gaussian conditioning in Hilbert spaces.

Item Type: Journal Article
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Science > Engineering
Faculty of Science > Mathematics
Library of Congress Subject Headings (LCSH): Conditional expectations (Mathematics), Hilbert space, Kernel functions, Machine learning -- Mathematical models, Gaussian measures
Journal or Publication Title: SIAM Journal on Mathematics of Data Science
Publisher: SIAM
ISSN: 2577-0187
Official Date: 13 July 2020
Dates:
DateEvent
13 July 2020Published
1 April 2020Accepted
Date of first compliant deposit: 14 August 2020
Volume: 2
Number: 3
Page Range: pp. 583-606
DOI: 10.1137/19M1305069
Status: Peer Reviewed
Publication Status: Published
Publisher Statement: “First Published in SIAM Journal on Mathematics of Data Science in 2(3), 2020 published by the Society for Industrial and Applied Mathematics (SIAM)” and the copyright notice as stated in the article itself (e.g., “Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.”)
Access rights to Published version: Open Access
Copyright Holders: © 2020, Society for Industrial and Applied Mathematics
RIOXX Funder/Project Grant:
Project/Grant IDRIOXX Funder NameFunder ID
UNSPECIFIED[DFG] Deutsche Forschungsgemeinschafthttp://dx.doi.org/10.13039/501100001659
390685689Konrad-Zuse-Zentrum für Informationstechnik Berlinhttp://viaf.org/viaf/133233189

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