# A Rigorous Theory of Conditional Mean Embeddings

@article{Klebanov2020ART,
title={A Rigorous Theory of Conditional Mean Embeddings},
author={Ilja Klebanov and Ingmar Schuster and Timothy John Sullivan},
journal={SIAM J. Math. Data Sci.},
year={2020},
volume={2},
pages={583-606}
}
• Published 2 December 2019
• Computer Science, Mathematics
• SIAM J. Math. Data Sci.
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 ...
14 Citations

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ILJA KLEBANOV1,*, BJÖRN SPRUNGK2,‡ and T. J. SULLIVAN1,3,† 1Zuse Institute Berlin, Takustraße 7, 14195 Berlin, Germany, E-mail: *klebanov@zib.de; †t.j.sullivan@warwick.ac.uk 2Technische Universität
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## References

SHOWING 1-10 OF 69 REFERENCES
A Measure-Theoretic Approach to Kernel Conditional Mean Embeddings
• Computer Science, Mathematics
NeurIPS
• 2020
A new operator-free, measure-theoretic definition of the conditional mean embedding as a random variable taking values in a reproducing kernel Hilbert space is presented, and a thorough analysis of its properties, including universal consistency is provided.
On the relation between universality, characteristic kernels and RKHS embedding of measures
• Mathematics, Computer Science
AISTATS
• 2010
The main contribution of this paper is to clarify the relation between universal and characteristic kernels by presenting a unifying study relating them to RKHS embedding of measures, in addition to clarifying their relation to other common notions of strictly pd, conditionally strictly pD and integrally strictlypd kernels.
Hilbert space embeddings of conditional distributions with applications to dynamical systems
• Mathematics, Computer Science
ICML '09
• 2009
This paper derives a kernel estimate for the conditional embedding, and shows its connection to ordinary embeddings, and aims to derive a nonparametric method for modeling dynamical systems where the belief state of the system is maintained as a conditional embeddedding.
Dimensionality Reduction for Supervised Learning with Reproducing Kernel Hilbert Spaces
• Mathematics
• 2004
We propose a novel method of dimensionality reduction for supervised learning problems. Given a regression or classification problem in which we wish to predict a response variable Y from an
Nonparametric Bayesian Inference with Kernel Mean Embedding
Kernel methods have been successfully used in many machine learning problems with favorable performance in extracting nonlinear structure of high-dimensional data. Recently, nonparametric inference
Foundations of modern probability
* Measure Theory-Basic Notions * Measure Theory-Key Results * Processes, Distributions, and Independence * Random Sequences, Series, and Averages * Characteristic Functions and Classical Limit
Vector Measures, volume 15 of Mathematical Surveys
• American Mathematical Society, Providence,
• 1977
Almost sure convergence of the largest and smallest eigenvalues of high-dimensional sample correlation matrices
• Mathematics
Stochastic Processes and their Applications
• 2018
In this paper, we show that the largest and smallest eigenvalues of a sample correlation matrix stemming from n independent observations of a p-dimensional time series with iid components converge
Theory of Reproducing Kernels and Applications, volume 44 of Developments in Mathematics
• 2016
Kernel Bayes' rule: Bayesian inference with positive definite kernels
• Mathematics, Computer Science
J. Mach. Learn. Res.
• 2013
A kernel method for realizing Bayes' rule is proposed, based on representations of probabilities in reproducing kernel Hilbert spaces, including Bayesian computation without likelihood and filtering with a nonparametric state-space model.