Neighborhood preserving embedding

TLDR

Geometric methods for high‑dimensional data analysis have gained traction, and unlike PCA which preserves global Euclidean structure, NPE focuses on maintaining local neighborhood relationships on the data manifold. The paper proposes a new subspace learning algorithm, neighborhood preserving embedding (NPE). NPE learns a subspace that preserves local neighborhoods of data sampled from a distribution supported near a Euclidean submanifold, and can be applied either in the original space or mapped into a reproducing kernel Hilbert space. Experiments on a face database show that NPE, which is less sensitive to outliers than PCA, is defined everywhere unlike Isomap and locally linear embedding, and its kernel extension further improves performance.

Abstract

Recently there has been a lot of interest in geometrically motivated approaches to data analysis in high dimensional spaces. We consider the case where data is drawn from sampling a probability distribution that has support on or near a submanifold of Euclidean space. In this paper, we propose a novel subspace learning algorithm called neighborhood preserving embedding (NPE). Different from principal component analysis (PCA) which aims at preserving the global Euclidean structure, NPE aims at preserving the local neighborhood structure on the data manifold. Therefore, NPE is less sensitive to outliers than PCA. Also, comparing to the recently proposed manifold learning algorithms such as Isomap and locally linear embedding, NPE is defined everywhere, rather than only on the training data points. Furthermore, NPE may be conducted in the original space or in the reproducing kernel Hilbert space into which data points are mapped. This gives rise to kernel NPE. Several experiments on face database demonstrate the effectiveness of our algorithm.