The separate tasks of denoising, conditional expectation and manifold
learning can often be posed in a common setting of finding the conditional
expectations arising from a product of two random variables. This paper focuses
on this more general problem and describes an operator theoretic approach to
estimating the conditional expectation. Kernel integral operators are used as a
compactification tool, to set up the estimation problem as a linear inverse
problem in a reproducing kernel Hilbert space. This equation is shown to have
solutions that are stable to numerical approximation, thus guaranteeing the
convergence of data-driven implementations. The overall technique is easy to
implement, and their successful application to some real-world problems are
also shown.

本文针对难题进行了运算符理论的研究与探讨，提出了套用了核积分算子的方法，能够有效地在复现核希尔伯特空间里解决线性逆问题的方法。

利用紧致核进行条件期望计算

Conditional expectation via compact kernels

Motivated by applications, we consider here new operator theoretic approaches
to Conditional mean embeddings (CME). Our present results combine a spectral
analysis-based optimization scheme with the use of kernels, stochastic
processes, and constructive learning algorithms. For initially given non-linear
data, we consider optimization-based feature selections. This entails the use
of convex sets of positive definite (p.d.) kernels in a construction of optimal
feature selection via regression algorithms from learning models. Thus, with
initial inputs of training data (for a suitable learning algorithm,) each
choice of p.d. kernel $K$ in turn yields a variety of Hilbert spaces and
realizations of features. A novel idea here is that we shall allow an
optimization over selected sets of kernels $K$ from a convex set $C$ of
positive definite kernels $K$. Hence our \textquotedblleft
optimal\textquotedblright{} choices of feature representations will depend on a
secondary optimization over p.d. kernels $K$ within a specified convex set $C$.

运用新的算子理论方法，结合核函数、随机过程和建设性学习算法，我们提出了用于条件均值嵌入的基于谱分析的优化方案，在优化模型特征选择的过程中，利用正定核的凸集。

通过正定核的条件均值嵌入和最优特征选择

Conditional mean embeddings and optimal feature selection via positive  definite kernels

"Masked Autoencoders (MAE) Are Scalable Vision Learners" revolutionizes the
self-supervised learning method in that it not only achieves the
state-of-the-art for image pre-training, but is also a milestone that bridges
the gap between visual and linguistic masked autoencoding (BERT-style)
pre-trainings. However, to our knowledge, to date there are no theoretical
perspectives to explain the powerful expressivity of MAE. In this paper, we,
for the first time, propose a unified theoretical framework that provides a
mathematical understanding for MAE. Specifically, we explain the patch-based
attention approaches of MAE using an integral kernel under a non-overlapping
domain decomposition setting. To help the research community to further
comprehend the main reasons of the great success of MAE, based on our
framework, we pose five questions and answer them with mathematical rigor using
insights from operator theory.

本文第一次提出了一种统一的理论框架，用数学方法解释 MAE 的强大表现，说明其基于补丁的注意力方法，并使用操作符理论阐明了 MAE 成功的主要原因。