Relative Distance-Based Laplacian Eigenmaps

In many areas of pattern recognition and machine learning, low dimensional data are often embedded in a high dimensional space. There have been many dimensionality reduction and manifold learning methods to discover the low dimensional representation from high dimensional data. Locality based manifold learning methods often rely on a distance metric between neighboring points. In this paper, we propose a new distance metric named relative distance, which is learned from the data and can better reflect the relative density. Combining the relative distance with Laplacian Eigenmaps (LE), we obtain a new algorithm called Relative Distance-based Laplacian Eigenmaps (RDLE) for nonlinear dimensionality reduction. Based on two different definitions of the relative distance, we give two variations of the RDLE. For efficient projection of out-of-sample data, we also present the linear version of RDLE, LRDLE. Experimental results on toy problems and real-world data demonstrate the effectiveness of our methods.