Recently, some matrix exponential-based discriminant analysis methods were proposed for high dimensionality reduction. It has been shown that they often have more discriminant power than the corresponding discriminant analysis methods. However, one has to solve some large-scale matrix exponential eigenvalue problems which constitutes the bottleneck in this type of methods. The main contribution of this paper is twofold: First, we propose a framework of fast implementation on general matrix exponential-based discriminant analysis methods. The key is to equivalently transform large-scale matrix computation problems into much smaller ones. On the other hand, it was mentioned in Wang et al. (IEEE Trans Image Process 23:920–930, 2014) that the exponential model is more reliable than the original one and suppresses the sensitivity to pertubations. However, the interpretation is only heuristic, and to the best of our knowledge, there is no theoretical justification for reliability and stability of the matrix discriminant analysis methods. To fill in this gap, the second contribution of our work is to provide stability analysis for the fast exponential discriminant analysis method from a theoretical point of view. Numerical experiments illustrate the numerical behavior of the proposed algorithm, and demonstrate that our algorithm is more stable than many state-of-the-art algorithms for high dimensionality reduction.

中文翻译：

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关于高维约简的一般矩阵指数判别分析方法

最近，提出了一些基于矩阵指数的判别分析方法以实现高维降维。已经表明，它们通常比相应的判别分析方法具有更大的判别能力。但是，必须解决一些大型矩阵指数特征值问题，这些问题构成了这类方法的瓶颈。本文的主要贡献有两个方面：首先，我们提出了一种基于通用矩阵指数的判别分析方法的快速实现框架。关键是将大型矩阵计算问题等效地转换为小得多的问题。另一方面，在Wang等人中提到过。（IEEE Trans Image Process 23：920–930，2014年），该指数模型比原始模型更可靠，并且抑制了对灌注的敏感性。但是，这种解释只是一种启发，就我们所知，矩阵判别分析方法的可靠性和稳定性没有理论上的依据。为了填补这一空白，我们的工作的第二个贡献是从理论上为快速指数判别分析方法提供稳定性分析。数值实验说明了该算法的数值行为，并证明了我们的算法比许多最新的高维降阶算法更稳定。我们工作的第二个贡献是从理论上为快速指数判别分析方法提供稳定性分析。数值实验说明了该算法的数值行为，并证明了我们的算法比许多最新的高维降阶算法更稳定。我们工作的第二个贡献是从理论上为快速指数判别分析方法提供稳定性分析。数值实验说明了该算法的数值行为，并证明了我们的算法比许多最新的高维降阶算法更稳定。