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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
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 | |||||||||
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| Subjects: | Q Science > QA Mathematics | |||||||||
| Divisions: | Faculty of Science > Engineering Faculty of Science > Mathematics |
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| 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: |
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| 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: |
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