Neighborhood Preserving Nonnegative Matrix Factorization

Nonnegative Matrix Factorization (NMF) has been widely used in computer vision and pattern recognition. It aims to find two nonnegative matrices whose product can well approximate the nonnegative data matrix, which naturally leads to parts-based and non-subtractive representation. In this paper, we present a neighborhood preserving nonnegative matrix factorization (NPNMF) for dimensionality reduction. It imposes an additional constraint on NMF that each data point can be represented as a linear combination of its neighbors. This constraint preserves the local geometric structure, and is good at dimensionality reduction on manifold. An iterative multiplicative updating algorithm is proposed to optimize the objective, and its convergence is guaranteed theoretically. Experiments on benchmark face recognition data sets demonstrate that the proposed method outperforms NMF as well as many state of the art dimensionality reduction methods.