Exact Subspace Clustering in Linear Time

Subspace clustering is an important unsupervised learning problem with wide applications in computer vision and data analysis. However, the state-of-the-art methods for this problem suffer from high time complexity---quadratic or cubic in $n$ (the number of data instances). In this paper we exploit a data selection algorithm to speedup computation and the robust principal component analysis to strengthen robustness. Accordingly, we devise a scalable and robust subspace clustering method which costs time only linear in $n$. We prove theoretically that under certain mild assumptions our method solves the subspace clustering problem exactly even for grossly corrupted data. Our algorithm is based on very simple ideas, yet it is the only linear time algorithm with noiseless or noisy recovery guarantee. Finally, empirical results verify our theoretical analysis

Large-Scale Hierarchical Classification via Stochastic Perceptron

Hierarchical classification (HC) plays an significant role in machine learning and data mining. However, most of the state-of-the-art HC algorithms suffer from high computational costs. To improve the performance of solving, we propose a stochastic perceptron (SP) algorithm in the large margin framework. In particular, a stochastic choice procedure is devised to decide the direction of next iteration. We prove that after finite iterations the SP algorithm yields a sub-optimal solution with high probability if the input instances are separable. For large-scale and high-dimensional data sets, we reform SP to the kernel version (KSP), which dramatically reduces the memory space needed. The KSP algorithm has the merit of low space complexity as well as low time complexity. The experimental results show that our KSP approach achieves almost the same accuracy as the contemporary algorithms on the real-world data sets, but with much less CPU running time