A Hilbert space embedding of a distribution--in short, a kernel mean embedding--has recently emerged as a powerful tool for machine learning and statistical 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. In addition to the classical applications of kernel methods, the kernel mean embedding has found novel applications in fields ranging from probabilistic modeling to statistical inference, causal discovery, and deep learning. This survey aims to give a comprehensive review of existing work and recent advances in this research area, and to discuss 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, 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 this new representation of distributions. We then discuss relationships between this framework and other related areas. Lastly, we give some suggestions on future research directions.

Provides a comprehensive review of kernel mean embeddings of distributions and, in the course of doing so, discusses some challenging issues that could potentially lead to new research directions. The targeted audience includes graduate students and researchers in machine learning and statistics.

This book constitutes the refereed proceedings of the 18th International Conference on Algorithmic Learning Theory, ALT 2007, held in Sendai, Japan, October 1-4, 2007, co-located with the 10th International Conference on Discovery Science, DS 2007. The 25 revised full papers presented together with the abstracts of five invited papers were carefully reviewed and selected from 50 submissions. They are dedicated to the theoretical foundations of machine learning.

The book covers theoretical questions including the latest extension of the formalism, and computational issues and focuses on some of the more fruitful and promising applications, including statistical signal processing, nonparametric curve estimation, random measures, limit theorems, learning theory and some applications at the fringe between Statistics and Approximation Theory. It is geared to graduate students in Statistics, Mathematics or Engineering, or to scientists with an equivalent level.

An advanced book for researchers and graduate students working in machine learning and statistics who want to learn about deep learning, Bayesian inference, generative models, and decision making under uncertainty. An advanced counterpart to Probabilistic Machine Learning: An Introduction, this high-level textbook provides researchers and graduate students detailed coverage of cutting-edge topics in machine learning, including deep generative modeling, graphical models, Bayesian inference, reinforcement learning, and causality. This volume puts deep learning into a larger statistical context and unifies approaches based on deep learning with ones based on probabilistic modeling and inference. With contributions from top scientists and domain experts from places such as Google, DeepMind, Amazon, Purdue University, NYU, and the University of Washington, this rigorous book is essential to understanding the vital issues in machine learning. Covers generation of high dimensional outputs, such as images, text, and graphs Discusses methods for discovering insights about data, based on latent variable models Considers training and testing under different distributions Explores how to use probabilistic models and inference for causal inference and decision making Features online Python code accompaniment

An integrated package of powerful probabilistic tools and key applications in modern mathematical data science.

This book reviews the state of the art in algorithmic approaches addressing the practical challenges that arise with hyperspectral image analysis tasks, with a focus on emerging trends in machine learning and image processing/understanding. It presents advances in deep learning, multiple instance learning, sparse representation based learning, low-dimensional manifold models, anomalous change detection, target recognition, sensor fusion and super-resolution for robust multispectral and hyperspectral image understanding. It presents research from leading international experts who have made foundational contributions in these areas. The book covers a diverse array of applications of multispectral/hyperspectral imagery in the context of these algorithms, including remote sensing, face recognition and biomedicine. This book would be particularly beneficial to graduate students and researchers who are taking advanced courses in (or are working in) the areas of image analysis, machine learning and remote sensing with multi-channel optical imagery. Researchers and professionals in academia and industry working in areas such as electrical engineering, civil and environmental engineering, geosciences and biomedical image processing, who work with multi-channel optical data will find this book useful.