In this paper, we introduce a higher order approach for dimension reduction based on the Trace Ratio problem. We show the existence and uniqueness of the solution, and we provide a relationship between the Trace Ratio problem and the Ratio Trace problem. We also propose a new algorithm to solve the Trace Ratio problem. We apply the approach to generalize the Linear Discriminant Analysis (LDA) to higher order tensors. We provide some numerical experiments to illustrate the efficiency of the proposed method. The method is based on the Einstein product, and it is a generalization of the state-of-the-art trace based DR methods to higher order tensors. The superiority of the Tensor-based methods have been shown experimentally, which motivates us to extend the state-of-the-art Ratio Trace based DR methods to higher order tensors via the Einstein product.

翻译：本文提出了一种基于迹比问题的高阶降维方法。我们证明了该问题解的存在性与唯一性，并建立了迹比问题与比迹问题之间的关系。同时，我们提出了一种求解迹比问题的新算法。通过将该方法应用于线性判别分析（LDA），我们将其推广至高阶张量形式。我们通过数值实验验证了所提方法的有效性。该方法基于爱因斯坦积，将当前最先进的基于迹的降维方法推广至高阶张量。实验结果表明基于张量的方法具有优越性，这促使我们通过爱因斯坦积将当前最先进的基于比迹的降维方法扩展至高阶张量。