This paper first proposes a statistic for measuring the correlation between two random variables. Because the data are usually polluted by noise and the feature of the data is usually task-relevant, this paper proposes to perform a transformation on the data before measuring their correlation. In addition, since the kernel method is a commonly used data transformation method in the field of machine learning, choosing different kernel functions is to choose different features, so we use kernel functions to perform this transformation. The random variable transformed by the kernel function becomes a kernelized random variable. Most importantly, the kernelized random variable is a random process, so we propose to use the norm of their cross-covariance function, which is called the kernelized cross-covariance criterion (kCCC), to measure the task-related correlation of two random variables. The kCCC criterion is a universal principle, based on which a variety of statistical machine learning algorithms can be constructed. This paper proposes to apply the kCCC to data dimensionality reduction, referred to as kCCC-DR for short. Further, we propose kCCC-DR in combination with the most widely studied and efficient local geometric property preservation method and manifold learning dimensionality reduction method, referred as kCCC-ML-DR for short. It is a dimensionality reduction method that maintains the global statistical characteristics and local geometric characteristics of the data at the same time. In the experiments presented in this paper, kCCC is combined with LLE, LE and LTSA. These algorithms are famous manifold learning algorithms, in which LLE is local linearity-preserving, LE is local similarity-preserving and LTSA is local homeomorphism-preserving. Experiments verify the effectiveness of our method.

本文首先提出了一个统计量来衡量两个随机变量之间的相关性。由于数据通常会受到噪声的污染，并且数据的特征通常与任务相关，因此本文提出在测量它们的相关性之前对数据进行转换。另外，由于核法是机器学习领域常用的数据变换方法，选择不同的核函数就是选择不同的特征，所以我们使用核函数来进行这种变换。核函数变换的随机变量成为核化的随机变量。最重要的是，核化的随机变量是一个随机过程，所以我们建议使用它们的互协方差函数的范数，称为核化的互协方差准则（kCCC)，来衡量两个随机变量的任务相关性。该kCCC标准是一个普遍的原则，在此基础上可以构建各种统计机器学习算法。本文提出将kCCC应用于数据降维，简称kCCC-DR。此外，我们提出了 kCCC-DR 结合研究最广泛和最有效的局部几何属性保存方法和流形学习降维方法，简称 kCCC-ML-DR。它是一种同时保持数据全局统计特征和局部几何特征的降维方法。在本文介绍的实验中，kCCC与 LLE、LE 和 LTSA 结合使用。这些算法都是著名的流形学习算法，其中LLE是局部线性保持，LE是局部相似性保持，LTSA是局部同胚保持。实验验证了我们方法的有效性。