Two-Dimensional Extensions of Neighborhood Preserving Embedding

Neighbourhood Preserving Embedding (NPE) is a novel subspace learning algorithm, which aims at preserving the local neighbourhood structure on the data manifold and is a linear approximation to Locally Linear Embedding (LLE). However, in typical image recognition in 1D vectors space, where the number of data samples is smaller than the dimension of data space, suffering from the singularity problem of matrix, NPE algorithm cannot be implemented directly. In this paper, we investigate NPE directly on image matrix for image recognition. The proposed two-dimensional neighbourhood preserving embedding (2DNPE) and bilateral two-dimensional neighbourhood preserving embedding (B2DNPE) algorithms are all based directly on 2D image matrices rather than on 1D vectors as NPE does, thus the problem of singularity confronted in 1D case is overcome. 2DNPE performs compression only in row direction, while B2DNPE performs compression both in row and in column direction. The relation of them to 2DLPP (B2DLPP) are also presented. The proposed algorithms are evaluated on ORL face database and handwritten digits database.