Conditional mean embeddings (CMEs) have proven themselves to be a powerful
tool in many machine learning applications. They allow the efficient
conditioning of probability distributions within the corresponding reproducing
kernel Hilbert spaces (RKHSs) by providing a linear-algebraic relation for the
kernel mean embeddings of the respective joint and conditional probability
distributions. Both centred and uncentred covariance operators have been used
to define CMEs in the existing literature. In this paper, we develop a
mathematically rigorous theory for both variants, discuss the merits and
problems of each, and significantly weaken the conditions for applicability of
CMEs. In the course of this, we demonstrate a beautiful connection to Gaussian
conditioning in Hilbert spaces.

本文通过对中心化和非中心化协方差算子 CME（Conditional mean embeddings） 的理论研究，讨论了 CME 的优点与问题，大幅放宽了 CME 的适用条件。在此过程中，本文演示了关于 Hilbert 空间中的高斯条件的美妙联系。

条件均值嵌入的严格理论

A Rigorous Theory of Conditional Mean Embeddings

We present a new methodology for sufficient dimension reduction (SDR). Our
methodology derives directly from the formulation of SDR in terms of the
conditional independence of the covariate $X$ from the response $Y$, given the
projection of $X$ on the central subspace [cf. J. Amer. Statist. Assoc. 86
(1991) 316--342 and Regression Graphics (1998) Wiley]. We show that this
conditional independence assertion can be characterized in terms of conditional
covariance operators on reproducing kernel Hilbert spaces and we show how this
characterization leads to an $M$-estimator for the central subspace. The
resulting estimator is shown to be consistent under weak conditions; in
particular, we do not have to impose linearity or ellipticity conditions of the
kinds that are generally invoked for SDR methods. We also present empirical
results showing that the new methodology is competitive in practice.

本文提出一种新的充分降维 (SDR) 的方法，该方法以协变量 $X$ 与响应 $Y$ 的条件独立性为基础，并以再生核希尔伯特空间上的条件协方差算子来表征该条件独立性断言，从而实现了对中心子空间的 $M$- 估计。