A Rigorous Theory of Conditional Mean Embeddings Article Swipe
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· 2020
· Open Access
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· DOI: https://doi.org/10.1137/19m1305069
· OA: W2991089358
Conditional mean embeddings (CMEs) have proven themselves to be a powerful\ntool in many machine learning applications. They allow the efficient\nconditioning of probability distributions within the corresponding reproducing\nkernel Hilbert spaces (RKHSs) by providing a linear-algebraic relation for the\nkernel mean embeddings of the respective joint and conditional probability\ndistributions. Both centred and uncentred covariance operators have been used\nto define CMEs in the existing literature. In this paper, we develop a\nmathematically rigorous theory for both variants, discuss the merits and\nproblems of each, and significantly weaken the conditions for applicability of\nCMEs. In the course of this, we demonstrate a beautiful connection to Gaussian\nconditioning in Hilbert spaces.\n