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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