Higher order multi-dimension reduction methods via Einstein product

TL;DR

This paper extends dimension reduction techniques to multi-dimensional data using the Einstein product, focusing on graph-based methods, and demonstrates their efficiency in high-dimensional applications like color images.

Contribution

It introduces novel multi-dimensional reduction methods via Einstein product, including variants with weights, and provides theoretical and experimental validation.

Findings

Enhanced performance on high-dimensional data

Effective handling of color images

Theoretical insights into generalized methods

Abstract

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.