Subspace clustering is a useful technique for many computer vision applications in which the intrinsic dimension of high-dimensional data is often smaller than the ambient dimension. Spectral clustering, as one of the main approaches to subspace clustering, often takes on a sparse representation or a low-rank representation to learn a block diagonal self-representation matrix for subspace generation. However, existing methods require solving a large scale convex optimization problem with a large set of data, with computational complexity reaches O(N^3) for N data points. Therefore, the efficiency and scalability of traditional spectral clustering methods can not be guaranteed for large scale datasets. In this paper, we propose a subspace clustering model based on the Kronecker product. Due to the property that the Kronecker product of a block diagonal matrix with any other matrix is still a block diagonal matrix, we can efficiently learn the representation matrix which is formed by the Kronecker product of k smaller matrices. By doing so, our model significantly reduces the computational complexity to O(kN^ 3 k ). Furthermore, our model is general in nature, and can be adapted to different regularization based subspace clustering methods. Experimental results on two public datasets show that our model significantly improves the efficiency compared with several state-of-the-art methods. Moreover, we have conducted experiments on synthetic data to verify the scalability of our model for large scale datasets.
Based on TRR, a 2D nonlinear variance regularized ridge regression (NVR3) @cite_28 was proposed to directly use 2D data, and thus the spatial information is maximally retained.
@cite_28
2736399023
Spectral clustering based subspace clustering methods have emerged recently. When the inputs are 2-dimensional (2D) data, most existing clustering methods convert such data to vectors as preprocessing, which severely damages spatial information of the data. In this paper, we propose a novel subspace clustering method for 2D data with enhanced capability of retaining spatial information for clustering. It seeks two projection matrices and simultaneously constructs a linear representation of the projected data, such that the sought projections help construct the most expressive representation with the most variational information. We regularize our method based on covariance matrices directly obtained from 2D data, which have much smaller size and are more computationally amiable. Moreover, to exploit nonlinear structures of the data, a nonlinear version is proposed, which constructs an adaptive manifold according to updated projections. The learning processes of projections, representation, and manifold thus mutually enhance each other, leading to a powerful data representation. Efficient optimization procedures are proposed, which generate non-increasing objective value sequence with theoretical convergence guarantee. Extensive experimental results confirm the effectiveness of proposed method.