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.

The Fisher linear discriminant analysis (LDA) is a classical method for classification and dimension reduction jointly. A major limitation of the conventional LDA is a so-called singularity issue. Many LDA variants, especially two-stage methods such as PCA+LDA and LDA/QR, were proposed to solve this issue. In the two-stage methods, an intermediate stage for dimension reduction is developed before the actual LDA method works. These two-stage methods are scalable because they are an approximate alternative of the LDA method. However, there is no theoretical analysis on how well they approximate the conventional LDA problem. In this paper we present theoretical analysis on the approximation error of a two-stage algorithm. Accordingly, we develop a new two-stage algorithm. Furthermore, we resort to a random projection approach, making our algorithm scalable. We also provide an implemention on distributed system to handle large scale problems. Our algorithm takes LDA/QR as its special case, and outperforms PCA+LDA while having a similar scalability. We also generalize our algorithm to kernel discriminant analysis, a nonlinear version of the classical LDA. Extensive experiments show that our algorithms outperform PCA+LDA and have a similar scalability with it.

In this paper we study sparsity-inducing nonconvex penalty functions using L´evy processes. We define such a penalty as the Laplace exponent of a subordina- tor. Accordingly, we propose a novel approach for the construction of sparsity- inducing nonconvex penalties. Particularly, we show that the nonconvex logarith- mic (LOG) and exponential (EXP) penalty functions are the Laplace exponents of Gamma and compound Poisson subordinators, respectively. Additionally, we explore the concave conjugate of nonconvex penalties. We find that the LOG and EXP penalties are the concave conjugates of negative Kullback-Leiber (KL) dis- tance functions. Furthermore, the relationship between these two penalties is due to asymmetricity of the KL distance.