Bayesian Low-Rank and Sparse Nonlinear Representation for Manifold Clustering

Abstract

Linear representation usually used by the optimization model about low-rankness and sparsity limits their applications to some extent. In this paper, we propose Bayesian low-rank and sparse nonlinear representation (BLSN) model exploiting nonlinear representation. Different from the optimization model, BLSN can be solved by traditional algorithm in Bayesian statistics easily without knowing the explicit mapping by kernel trick. Moreover, it can learn the parameters adaptively to choose the low-rank and sparse properties and also provides a way to enforce more properties on one quantity in a Bayesian model. Based on the observation that the data points drawn from a union of manifolds may gain more meaningful linear structure after a nonlinear mapping, we apply BLSN for manifold clustering. It can handle different problems by constructing various kernels. With respect to the case of linear manifold, known as subspace segmentation, we propose a kernel by the Veronese mapping. In addition, we also design the kernel matrices for the case of nonlinear manifold. Experimental results confirm the effectiveness and the potential of our model for manifold clustering.