Locality preserving projection (LPP) is a widely used linear dimensionality reduction method, which preserves the locality structure of the original data. Motivated by the fact that kernel technique can capture nonlinear similarity of features and help to improve separability between nearby data points, this paper proposes locality preserving projection model based on Euler representation (named as ELPP). This model first projects the data into a complex space with Euler representation, then learns the dimensionality reduction projection with preserving locality structure in this complex space. We also extend ELPP to F-ELPP by replacing the squared F-norm with F-norm, which will weaken the exaggerated errors and be more robustness to outliers. The optimization algorithms of the two models are given, and the convergence of F-ELPP is proved. A large number of experiments on several public databases have demonstrated that the two proposed models have good robustness and feature extraction ability.

局部性保留投影（LPP）是一种广泛使用的线性降维方法，用于保留原始数据的局部性结构。基于核技术可以捕获特征的非线性相似性并有助于改善附近数据点之间的可分离性这一事实，本文提出了一种基于欧拉表示的局部保留投影模型（称为ELPP）。该模型首先将数据投影到具有Euler表示的复杂空间中，然后学习在该复杂空间中保留局部结构的降维投影。通过将平方F范数替换为F范数，我们还将ELPP扩展到F-ELPP，这将减弱夸大的误差并提高对异常值的鲁棒性。给出了两个模型的优化算法，并证明了F-ELPP的收敛性。