Kernel Mean Embedding of Distributions: A Review and Beyond (1605.09522v4)

Abstract: 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.

• The paper provides a comprehensive review of kernel mean embedding, demonstrating how to map probability distributions into RKHS for enhanced inference and learning.
• It details the use of characteristic kernels that ensure injectivity, alongside theoretical guarantees and effective strategies for empirical estimation.
• Key applications include two-sample testing, conditional mean embedding, and scalable algorithms in machine learning, with promising directions for high-dimensional data and causal discovery.

Overview of "Kernel Mean Embedding of Distributions: A Review and Beyond"

The paper "Kernel Mean Embedding of Distributions: A Review and Beyond" provides an exhaustive review and analysis of the concept of kernel mean embedding, a methodology that maps probability distributions into reproducing kernel Hilbert spaces (RKHS). This embedding allows the application of kernel methods, traditionally used for data points, to entire distributions, significantly extending their applicability across machine learning and statistical domains.

Key Concepts

1. Kernel Mean Embedding: This technique generalizes the feature map used in support vector machines (SVMs) by embedding probability distributions into an RKHS. The embedding is characterized by a positive definite kernel, enabling detailed representation of distributions within the constraints of the RKHS specific to the kernel used.
2. Hilbert Space Representation: The paper details how distributions can be represented in a Hilbert space via kernel mean embeddings, providing essential insights into theoretical aspects like mapping invariance and properties of the embedded space.
3. Empirical Estimation and Properties: Practical aspects of estimating these embeddings from sample data are discussed alongside theoretical guarantees such as convergence properties and the nature of characteristic kernels, which ensure injectivity of the mapping from distributions to the RKHS.
4. Applications in Statistical Inference and Machine Learning: The embedding framework facilitates non-parametric Bayesian inference, hypothesis testing, causal inference, and learning on data structures such as graphs and sequences. Examples include two-sample testing through Maximum Mean Discrepancy (MMD) and independence testing via Hilbert-Schmidt Independence Criterion (HSIC).
5. Conditional Mean Embedding: This extends the concept to conditional distributions, allowing for operations like inference in graphical models, probabilistic programming, and reinforcement learning in a non-parametric setting. The survey explores implications for the sum and product rules fundamental to probabilistic reasoning.

Implications and Future Directions

• Scalability and High-Dimensional Data: The necessity for efficient computation of kernel mean embeddings in big data contexts is underscored, alongside challenges posed by high-dimensional data distributions. The paper encourages further exploration of approximation methods and scalable algorithms.
• Causal Discovery: There's a focus on the utility of kernel mean embeddings in uncovering causal relationships, an area ripe for future research given the intersection with modern causal inference techniques.
• Privacy and Distributional Learning: Embedding privacy-preserving mechanisms into the embedding process and expanding distributional learning paradigms are highlighted as promising directions.
• Interdisciplinary Applications: The versatility of the kernel mean embedding methodology suggests applications in various fields like bioinformatics, computer vision, and beyond, inviting interdisciplinary applications and novel algorithmic improvements.

Conclusion

The paper provides a foundational yet comprehensive review of the kernel mean embedding approach, consolidating its theoretical infrastructure and demonstrating its broad potential across machine learning and statistics. By mapping distributions into RKHS, computational and theoretical tools of kernel methods can be leveraged to tackle complex inference and learning tasks in innovative ways. The survey sets the stage for advancing this methodology further, inspiring new research, applications, and theoretical insights in AI and beyond.