Local Tangent Space Alignment

Abstract

In this chapter, a dimensionality reduction embedding method, called local tangent space alignment (LTSA), is introduced. The method is based on the same geometric intuitions as LLE: If a data set is sampled from a smooth manifold, then the neighbors of each point remain nearby and similarly co-located in the low dimensional space. LTSA uses a different approach to the embedded space compared with LLE. In LLE, each point in the data set is linearly embedded into a locally linear patch of the manifold. Then low-dimensional data are constructed so that the locally linear relations of the original data are preserved. In LTSA, a locally linear patch is constructed by applying PCA on the neighbors. The patch then can be considered as an approximation of the tangent space at the point. Since the tangent place provides a coordinate representation of the neighbors, the coordinates give a low-dimensional representation of the patch. An alignment technique is introduced in LTSA to align the local representation to a global one. The chapter is organized as follows. In Section 11.1, we describe the method, paying more attention to the global alignment technique. In Section 11.2, the LTSA algorithm is developed. In Section 11.3, the experiments of LTSA are presented.