Locality preserving projections (LPP) is a popular unsupervised dimensionality reduction method based on manifold learning. As a supervised version of the LPP method, discriminant locality preserving projections (DLPP) method has been recently proposed and paid much attention to by researchers. However, the DLPP method has the small-sample-size (SSS) problem. In this paper, in the view of the eigenvalues of scattering matrices of DLPP, they are first mapped to the new values by two polynomial functions, and with the properties of the matrix function of the two polynomial functions, the criterion of the DLPP method is reconstructed; thus, a novel dimensionality reduction method, named polynomial discriminant locality preserving projections (PDLPP) method, is proposed. The proposed PDLPP method has two advantages: one is that it addresses the SSS problem of DLPP, and the other is that, with the nonlinear mapping implied by PDLPP, the distance between inter-class samples is much enlarged and then the better performance of pattern classification is achieved. The experiments are conducted on the COIL-20 database, ORL, Georgia Tech, and AR face datasets, and the results show that the PDLPP is superior to state-of-the-art methods.

局部性保留投影（LPP）是一种基于流形学习的流行的无监督降维方法。作为LPP方法的一种监督版本，最近提出了判别局部性保留投影（DLPP）方法，并且受到了研究者的广泛关注。但是，DLPP方法存在小样本大小（SSS）问题。本文针对DLPP散射矩阵的特征值，首先通过两个多项式函数将其映射为新值，并利用两个多项式函数的矩阵函数的性质，将DLPP方法的准则为重建 因此，提出了一种新的降维方法，即多项式判别局部性保留投影（PDLPP）方法。提出的PDLPP方法具有两个优点：一是解决了DLPP的SSS问题，二是通过PDLPP隐含的非线性映射，极大地扩大了类间样本之间的距离，从而实现了较好的模式分类性能。实验在COIL-20数据库，ORL，乔治亚理工学院和AR人脸数据集上进行，结果表明PDLPP优于最新方法。