基于欧拉表示的线性判别分析

Linear Discriminant Analysis Based on Euler Representation

摘要: 线性判别分析(LDA)是一种经典的线性降维方法，广泛应用于统计学和机器学习领域，用于提取数据中的判别特征。然而，随着数据规模的迅速扩大，大量数据呈现出复杂的非线性特征，使得传统依赖欧氏距离的LDA在处理此类数据时难以充分捕捉复杂的数据结构。为了解决这一问题，本文提出了一种基于欧拉表示的线性判别分析模型(Euler-LDA)。Euler-LDA结合欧拉表示的优势，将数据映射到复数空间。通过利用复数的几何特性，更精确地捕捉数据的非线性关系，从而显著提高算法在复杂数据分布场景下的适用性和分类性能。此外，这种方法在处理复杂数据分布时，展现出较强的适用性与优越的分类性能。此外，通过进一步拉大类间距离与缩小类内距离，Euler-LDA有效提升了特征提取的准确性。通过这些机制，欧拉表示能够更加高效地应对非线性分布的数据，提供比传统欧氏距离更可靠和精确的相似性度量。在多个数据库上的对比实验结果表明，该算法识别率显著优于传统LDA及其改进方法。

Abstract: Linear Discriminant Analysis (LDA) is a classic linear dimensionality reduction method widely used in statistics and machine learning for extracting discriminative features from data. However, as data scales rapidly expand, a large amount of data exhibits complex nonlinear characteristics, making it difficult for traditional LDA, which relies on Euclidean distance, to fully capture intricate data structures when processing such data. To address this issue, this paper proposes a Linear Discriminant Analysis model based on Euler representation (Euler-LDA). Euler-LDA leverages the advantages of Euler representation to map data into a complex space. By utilizing the geometric properties of complex numbers, it more accurately captures the nonlinear relationships in the data, thereby significantly improving the algorithm’s applicability and classification performance in scenarios with complex data distributions. Furthermore, this method demonstrates strong applicability and superior classification performance when dealing with complex data distributions. Additionally, by further increasing the inter-class distance and reducing the intra-class distance, Euler-LDA effectively enhances the accuracy of feature extraction. Through these mechanisms, Euler representation can more efficiently handle nonlinearly distributed data, providing a more reliable and precise similarity measure than traditional Euclidean distance. Comparative experimental results on multiple databases show that the recognition rate of this algorithm is significantly superior to that of traditional LDA and its improved methods.