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Kernel Mean Embedding of Distributions: A Review and Beyonds

108 pages of goodness !

Kernel Mean Embedding of Distributions: A Review and Beyonds by

Krikamol Muandet,

Kenji Fukumizu,

Bharath Sriperumbudur,

Bernhard Schölkopf
A Hilbert space embedding of distributions---in short, kernel mean
embedding---has recently emerged as a powerful machinery for probabilistic
modeling, statistical inference, machine learning, and causal discovery. The
basic idea behind this framework is to map distributions into a reproducing
kernel Hilbert space (RKHS) in which the whole arsenal of kernel methods can be
extended to probability measures. It gave rise to a great deal of research and
novel applications of positive definite kernels. The goal of this survey is to
give a comprehensive review of existing works and recent advances in this
research area, and to discuss some of the most challenging issues and open
problems that could potentially lead to new research directions. The survey
begins with a brief introduction to the RKHS and positive definite kernels
which forms the backbone of this survey, followed by a thorough discussion of
the Hilbert space embedding of marginal distributions, theoretical guarantees,
and review of its applications. The embedding of distributions enables us to
apply RKHS methods to probability measures which prompts a wide range of
applications such as kernel two-sample testing, independent testing, group
anomaly detection, and learning on distributional data. Next, we discuss the
Hilbert space embedding for conditional distributions, give theoretical
insights, and review some applications. The conditional mean embedding enables
us to perform sum, product, and Bayes' rules---which are ubiquitous in
graphical model, probabilistic inference, and reinforcement learning---in a
non-parametric way using the new representation of distributions in RKHS. We
then discuss relationships between this framework and other related areas.
Lastly, we give some suggestions on future research directions.

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