Isometric Projection

Abstract

Recently the problem of dimensionality reduction has received a lot of interests in many fields of information processing, including data mining, information retrieval, and pattern recognition. We consider the case where data is sampled from a low dimensional manifold which is embedded in high dimensional Euclidean space. The most popular manifold learning algorithms include Locally Linear Embedding, ISOMAP, and Laplacian Eigenmap. However, these algorithms are nonlinear and only provide the embedding results of training samples. In this paper, we propose a novel linear dimensionality reduction algorithm, called Isometric Projection. Isometric Projection constructs a weighted data graph where the weights are discrete approximations of the geodesic distances on the data manifold. A linear subspace is then obtained by preserving the pairwise distances. Our algorithm can be performed in either original space or reproducing kernel Hilbert space, which leads to Kernel Isometric Projection. In this way, Isometric Projection can be defined everywhere. Comparing to Principal Component Analysis (PCA) which is widely used in data processing, our algorithm is more capable of discovering the intrinsic geometrical structure. Specially, PCA is optimal only when the data space is linear, while our algorithm has no such assumption and therefore can handle more complex data space. We present experimental results of the algorithm applied to synthetic data set as well as real life data. These examples illustrate the effectiveness of the proposed method. 1