Multi-subspace representation and discovery

Abstract

Abstract. This paper presents the multi-subspace discovery problem and provides a theoretical solution which is guaranteed to recover the number of subspaces, the dimensions of each subspace, and the mem-bers of data points of each subspace simultaneously. We further propose a data representation model to handle noisy real world data. We de-velop a novel optimization approach to learn the presented model which is guaranteed to converge to global optimizers. As applications of our models, we first apply our solutions as preprocessing in a series of ma-chine learning problems, including clustering, classification, and semi-supervised learning. We found that our method automatically obtains robust data presentation which preserves the affine subspace structures of high dimensional data and generate more accurate results in the learn-ing tasks. We also establish a robust standalone classifier which directly utilizes our sparse and low rank representation model. Experimental re-sults indicate our methods improve the quality of data by preprocessing and the standalone classifier outperforms some state-of-the-art learning approaches.