Locality preserving projection (LPP), as a well-known technique for dimensionality reduction, is designed to preserve the local structure of the original samples which usually lie on a low-dimensional manifold in the real world. However, it suffers from the undersampled or small-sample-size problem, when the dimension of the features is larger than the number of samples which causes the corresponding generalized eigenvalue problem to be ill-posed. To address this problem, we show that LPP is equivalent to a multivariate linear regression under a mild condition, and establish the connection between LPP and a least squares problem with multiple columns on the right-hand side. Based on the developed connection, we propose two regularized least squares methods for solving LPP. Experimental results on real-world databases illustrate the performance of our methods.

局部性保留投影（LPP）是一种众所周知的降维技术，旨在保留原始样本的局部结构，该样本通常位于现实世界中的低维流形上。但是，当特征的维数大于样本数量时，会遭受欠采样或小样本大小的问题，这会使相应的广义特征值问题不适当地解决。为了解决这个问题，我们证明了LPP在温和的条件下等效于多元线性回归，并建立了LPP与最小二乘问题的联系，并在右侧有多个列。基于已开发的连接，我们提出了两种求解LPP的正则化最小二乘法。