Real local-linearity preserving embedding

In recent years, manifold learning methods have aroused a great interest in the machine learning community. A key issue that determines the effectiveness of the manifold learning methods is how to accurately capture the local geometry of the low-dimensional manifold. However, most of the manifold learning algorithms cannot exploit the real local geometry if the neighbors for each sample point are not correctly selected. In this paper, we address this problem in the context of locally linear embedding (LLE). A new local optimization model is proposed to find the local weights that can represent the real local manifold geometry when the neighborhoods contain wrong neighbors. A new algorithm called real local-linearity preserving embedding (RLLPE) is then proposed by preserving the exploited real local geometry. We demonstrate the improvement and efficiency of RLLPE using both synthetic and real-word data.