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Sparse subspace clustering (SSC), as one of the most successful subspace clustering methods, has achieved notable clustering accuracy in computer vision tasks. However, SSC applies only to vector data in Euclidean space. As such, there is still no satisfactory approach to solve subspace clustering by self-expressive principle for symmetric positive definite (SPD) matrices which is very useful in computer vision. In this paper, by embedding the SPD matrices into a Reproducing Kernel Hilbert Space (RKHS), a kernel subspace clustering method is constructed on the SPD manifold through an appropriate Log-Euclidean kernel, termed as kernel sparse subspace clustering on the SPD Riemannian manifold (KSSCR). By exploiting the intrinsic Riemannian geometry within data, KSSCR can effectively characterize the geodesic distance between SPD matrices to uncover the underlying subspace structure. Experimental results on two famous database demonstrate that the proposed method achieves better clustering results than the state-ofthe-art approaches.

Stone's method is one of the novel approaches to the blind source separation (BSS) problem and is based on Stone's conjecture. However, this conjecture has not been proved. We present a simple simulation to demonstrate that Stone's conjecture is incorrect. We then modify Stone's conjecture and prove this modified conjecture as a theorem, which can be used a basis for BSS algorithms. Problem FormulationBlind source separation (BSS) separates the source signals from the observed signals without any prior information of the source signals and the transfer channel. "Blind" means that both the source and the channel are unknown.The mathematical model of this problem is written aswhere equation 1.1 is the model of the mixture, while equation 1.2 is the model of the separation. The blind (unknown) source signal vector is denoted as S(t) = (s 1 (t), s 2 (t), · · · , s n (t)) T , and the observed signal vector is denoted as X(t) = (x 1 (t), x 2 (t), · · · , x m (t)) T . The vector Y(t) = (y 1 (t), y 2 (t), · · · , y n (t)) T indicating the separated signals. A is the unknown n × m mixture matrix (indicating the unknown channel). W is the separating matrix. The aim of BSS is to get the expressionby adjusting the separation matrix W. In equation 1.3, P is a permutation matrix, while D is a diagonal matrix. Usually the source signals are only assumed to be statistically independent.