Distortion-Free Nonlinear Dimensionality Reduction

Nonlinear Dimensionality Reduction is an important issue in many machine learning areas where essentially low-dimensional data is nonlinearly embedded in some high-dimensional space. In this paper, we show that the existing Laplacian Eigenmaps method suffers from the distortion problem, and propose a new distortion-free dimensionality reduction method by adopting a local linear model to encode the local information. We introduce a new loss function that can be seen as a different way to construct the Laplacian matrix, and a new way to impose scaling constraints under the local linear model. Better low-dimensional embeddings are obtained via constrained concave convex procedure. Empirical studies and real-world applications have shown the effectiveness of our method.