Global and local choice of the number of nearest neighbors in locally linear embedding

• Published in: Pattern recognition letters Vol. 32; no. 16; pp. 2171 - 2177
• Main Authors: Álvarez-Meza, Andrés, Valencia-Aguirre, Juliana, Daza-Santacoloma, Genaro, Castellanos-Domínguez, Germán
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01-12-2011
• Subjects: Dimensionality reduction; Embedding quality; Locally linear embedding; Number of nearest neighbors; Dimensionality reduction; Number of nearest neighbors; Locally linear embedding; Embedding quality
• ISSN: 0167-8655; 1872-7344
• DOI: 10.1016/j.patrec.2011.05.011

► We propose a new method for automatically computing the number of neighbors in LLE. ► We analyze the global and local properties of the embedding results. ► We study manifolds where the density and the intrinsic dimensionality of the neighborhoods are variable. ► Artificial and real-world datasets were tested.
The crux in the locally linear embedding algorithm is the selection of the number of nearest neighbors k. Some previous techniques have been developed for finding this parameter based on embedding quality measures. Nevertheless, they do not achieve suitable results when they are tested on several kind of manifolds. In this work is presented a new method for automatically computing the number of neighbors by means of analyzing global and local properties of the embedding results. Besides, it is also proposed a second strategy for choosing the parameter k, on manifolds where the density and the intrinsic dimensionality of the neighborhoods are changeful. The first proposed technique, called preservation neighborhood error, calculates a unique value of k for the whole manifold. Moreover, the second method, named local neighborhood selection, computes a suitable number of neighbors for each sample point in the manifold. The methodologies were tested on artificial and real-world datasets which allow us to visually confirm the quality of the embedding. According to the results our methods aim to find suitable values of k and appropriated embeddings.