This paper explores the extension of dimension reduction (DR) techniques to the multi-dimension case by using the Einstein product. Our focus lies on graph-based methods, encompassing both linear and nonlinear approaches, within both supervised and unsupervised learning paradigms. Additionally, we investigate variants such as repulsion graphs and kernel methods for linear approaches. Furthermore, we present two generalizations for each method, based on single or multiple weights. We demonstrate the straightforward nature of these generalizations and provide theoretical insights. Numerical experiments are conducted, and results are compared with original methods, highlighting the efficiency of our proposed methods, particularly in handling high-dimensional data such as color images.

中文翻译：

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通过爱因斯坦积的高阶多维约简方法

本文探讨了使用爱因斯坦乘积将降维（DR）技术扩展到多维情况。我们的重点在于基于图的方法，包括监督和无监督学习范式中的线性和非线性方法。此外，我们还研究了线性方法的斥力图和核方法等变体。此外，我们基于单个或多个权重对每种方法提出了两种概括。我们展示了这些概括的简单本质并提供了理论见解。进行了数值实验，并将结果与​​原始方法进行了比较，突出了我们提出的方法的效率，特别是在处理彩色图像等高维数据方面。