Geometric weighting subspace clustering on nonlinear manifolds

Considering that the conventional subspace clustering methods of sparse subspace clustering (SSC) and low-rank representation (LRR) are only applicable to linear manifolds, we propose a novel subspace clustering framework that generalizes them for nonlinear manifolds. To do this, we integrate a weighting matrix and kernel matrix into the regularization of this framework. The weighting matrix is calculated using the similarities between tangent spaces on data manifolds and the Euclidean distances between data points, so that it can explicitly characterize the intrinsic geometry of data manifolds. Besides, we provide a geometrical interpretation for the effects of weighted ℓ∗-norm involved in the proposed framework, exploiting symmetric gauge function (SGF) of von Neumann theory that establishes a relationship exactly between singular and matrix norm. To solve the regularization with respect to the weighted norm, we design a fixed-point continuation algorithm to obtain an approximate closed solution. Experimental results on three computer vision tasks show the superiority of clustering accuracy over other similar approaches and demonstrate the effectiveness of the weighting matrix. That also proves the proposed method has better interpretability than other state-of-the-art methods.