Low-rank preserving embedding

Abstract In this paper, we consider the problem of linear dimensionality reduction with the novel technique of low-rank representation, which is a promising tool of discovering subspace structures of given data. Existing approaches based on graph embedding usually capture structure of data via stacking the local structure of each datum, such as neighborhood graph, l1-graph and l2-graph. Yet they lack explicit discrimination between those local structures and suffer from corrupted samples. To this end, we propose a new linear dimensionality reduction method by virtue of the lowest rank representation (LRR) of data, which is dubbed low-rank preserving embedding (LRPE). Different from the traditional routes, LRPE achieves all data self-representations jointly and can thus extract the global structure of a data set as a whole. The global low-rank constraint explicitly enforces the LRR matrix to be block-diagonal form, so that the samples with a similar intrinsic structure, which are more likely to be from the same class, are described by a similar set of bases. Hence, LRPE is discriminative even if no class labels are provided. Benefiting from the robust LRR, LRPE is also robust to various noises and errors in data. Besides, we rewritten all related methods into a unified formulation, followed by a detailed solution and clear comparisons. Finally, we conduct extensive experiments on publicly available data sets for data visualization and classification. The inspiring experimental results show the effectiveness, the cheap computation and the robustness of the proposed method.