Graph-Regularized Fast Low-Rank Matrix Approximation Using The NystrÖM Method for Clustering

Abstract

In recovering low-dimensional representations of high-dimensional data, graph or manifold-regularized schemes have been investigated as a key tool in many areas to preserve the neighborhood structure of the data set. In spite of its effectiveness, these methods are often not tractable in practice, because graph structures of data lead to a large matrix (e.g., affinity of Laplacian matrix) and the methods require eigenanalysis of it interactively. In this paper, we propose an efficient low-rank matrix approximation that regularized by graph information derived from row and column range spaces of data. To deal with high computational complexity issue, we leverage the Nyström method, which has been universally used to approximate low-rank component of Symmetric Positive Semi-Definite (SPSD) matrices with sampling. Moreover, we devise a Clustered Nystrom extension with QR decomposition to efficiently aggregate more information from samples and to accurately approximate low-rank structure. We compare the performance of the proposed algorithm with other several general algorithms in clustering experiments on benchmark dataset. Our experimental results show that our method has a favorable running speed while the accuracy of our proposed method is better or comparable to the competing methods.