${L_{2,1}}$ -Norm Discriminant Manifold Learning

Abstract

Recently, ${L_{1}}$ -norm-based robust discriminant feature extraction technique has been attracted much attention in dimensionality reduction and pattern recognition. However, it does not relate to the scatter matrix which well characterizes the geometric structure of data. In this paper, we propose a robust formulation of graph embedding framework for dimensionality reduction. In this robust framework, we use ${L_{2}}$ -norm to measure the distance along space dimension and ${L_{1}}$ -norm to sum overall data points. The proposed robust graph embedding framework retains the traditional framework’s desirable properties, such as rotational invariance and well geometric structure, and simultaneously suppresses outliers. Based on this framework, we develop a simple and robust feature extraction method, namely ${L_{2,1}}$ -norm-based discriminant locality preserving projections ( ${L_{2,1}}$ -DLPP) and provide an effective iterative algorithm to solve ${L_{2,1}}$ -DLPP. Extensive experiments in artificial data and three popular face databases illustrate the effectiveness of our proposed method.