A Hilbert space embedding of a distribution---in short, a kernel mean embedding---has recently emerged as a powerful tool for machine learning and inference. 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 can be viewed as a generalization of the original "feature map" common to support vector machines (SVMs) and other kernel methods. While initially closely associated with the latter, it has meanwhile found application in fields ranging from kernel machines and probabilistic modeling to statistical inference, causal discovery, and deep learning. The goal of this survey is to give a comprehensive review of existing work and recent advances in this research area, and to discuss the most challenging issues and open problems that could 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 a 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, 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. We then discuss relationships between this framework and other related areas. Lastly, we give some suggestions on future research directions.Comment: 147 pages; this is a version of the manuscript after the review proces

Transfer operators such as the Perron-Frobenius or Koopman operator play an important role in the global analysis of complex dynamical systems. The eigenfunctions of these operators can be used to detect metastable sets, to project the dynamics onto the dominant slow processes, or to separate superimposed signals. We propose kernel transfer operators, which extend transfer operator theory to reproducing kernel Hilbert spaces and show that these operators are related to Hilbert space representations of conditional distributions, known as conditional mean embeddings. The proposed numerical methods to compute empirical estimates of these kernel transfer operators subsume existing data-driven methods for the approximation of transfer operators such as extended dynamic mode decomposition and its variants. One main benefit of the presented kernelbased approaches is that they can be applied to any domain where a similarity measure given by a kernel is available. Furthermore, we provide elementary results on eigendecompositions of finite-rank RKHS operators. We illustrate the results with the aid of guiding examples and highlight potential applications in molecular dynamics as well as video and text data analysis.

Figure 1: Ground truth grasps and generated grasps. Each row corresponds to one object. Left three columns show the ground truth grasps, each from three different viewpoints. The middle three columns show one generated example, and the right three columns show another generated example. Note that these objects are never seen during training. See Appendix E (Fig. E.4) for more examples.

A Hilbert space embedding of a distribution-in short, a kernel mean embedding-has recently emerged as a powerful tool for machine learning and inference. 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 can be viewed as a generalization of the original "feature map" common to support vector machines (SVMs) and other kernel methods. While initially closely associated with the latter, it has meanwhile found application in fields ranging from kernel machines and probabilistic modeling to statistical inference, causal discovery, and deep learning.The goal of this survey is to give a comprehensive review of existing work 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 a 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. The targeted audience includes graduate students and researchers in machine learning and statistics who are interested in the theory and applications of kernel mean embeddings.G over some input space Y, we havewhere U Y |x denotes the embedding of the conditional distribution P(Y |X = x). That is, we can compute a conditional expected value of any function g ∈ G w.r.t. P(Y |X = x) by taking an inner product in G between the function g and the embedding of P(Y |X = x) (see Section 4A Synopsis. As a result of the aforementioned advantages, the kernel mean embedding has made widespread contributions in various directions. Firstly, most tasks in machine learning and statistics involve estimation of the data-generating process whose success depends critically on the accuracy and the reliability of this estimation. It is known that estimating the kernel mean embedding is easier than estimating the distribution itself, which helps improve many statistical inference methods. These include, for example, two-sample testing (