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.

翻译：本文探讨了利用爱因斯坦积将降维技术推广至多维情形的方法。研究聚焦于基于图的线性与非线性方法，涵盖监督学习与非监督学习范式。此外，我们研究了线性方法的变体，包括排斥图与核方法。进一步地，针对每种方法，我们提出了基于单权重与多权重的两种推广形式，论证了此类推广的简洁性，并提供了理论洞见。通过数值实验并与原始方法结果对比，揭示了所提方法在处理高维数据（如彩色图像）时的有效性。