Locally Linear Embedding

Abstract

In this chapter, locally linear embedding (LLE) method for dimensionality reduction is introduced. The method is based on simple geometric intuitions: If a data set is sampled from a smooth manifold, then neighbors of each point remain nearby and are similarly co-located in the low-dimensional space. In LLE, each point in the data set is linearly embedded into a locally linear patch of the manifold. Then low-dimensional data is constructed such that the locally linear relations of the original data are preserved. The chapter is organized as follows. In Section 10.1, we geometrically describe LLE method and its algorithm. The experiments of LLE are presented in Section 10.2 and some applications are introduced in Section 10.3. The mathematical justification of LLE is given in Section 10.4.