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.
As pointed out in @cite_23 , SSC finds the sparsest representation of each data vector individually. There is no global constraint on its solution, so the SSC method may be inaccurate at capturing the global structures of data. Liu @cite_23 proposed that low rank can be a more appropriate criterion. Similar to SSC, LRR aims to find a low-rank representation of @math by solving the following optimization problem, since the nuclear norm @math is the best convex approximation of @math over the unit ball of matrices: where @math is the sum of the singular values of @math .
@cite_23
2008138666 2944309706 79405465 2554853703
Low-rank representation (LRR) intends to find the representation with lowest rank of a given data set, which can be formulated as a rank-minimisation problem. Since the rank operator is non-convex and discontinuous, most of the recent works use the nuclear norm as a convex relaxation. It is theoretically shown that, under some conditions, the Frobenius-norm-based optimisation problem has a unique solution that is also a solution of the original LRR optimisation problem. In other words, it is feasible to apply the Frobenius norm as a surrogate of the non-convex matrix rank function. This replacement will largely reduce the time costs for obtaining the lowest-rank solution. Experimental results show that the method (i.e. fast LRR (fLRR)) performs well in terms of accuracy and computation speed in image clustering and motion segmentation compared with nuclear-norm-based LRR algorithm. An important problem in analyzing big data is subspace clustering , i.e., to represent a collection of points in a high-dimensional space via the union of low-dimensional subspaces. Sparse subspace clustering (SSC) and Low-rank representation (LRR) are the state-of-the-art methods for this task. These two methods are fundamentally similar in that both are based on convex optimization exploiting the intuition of “Self-Expressiveness”. The main difference is that the SSC minimizes the vector @math norm of the representation matrix to induce sparsity while LRR minimizes the nuclear norm (aka trace norm) to promote a low-rank structure. Because the representation matrix is often simultaneously sparse and low-rank, we propose a new algorithm, termed Low-rank sparse subspace clustering (LRSSC), by the combining SSC and LRR, and develop theoretical guarantees of the success of the algorithm. The results reveal interesting insights into the strengths and the weaknesses of SSC and LRR, and demonstrate how the LRSSC can take advantage of both methods in preserving the “Self-Expressiveness Property” and “Graph Connectivity” at the same time. A byproduct of our analysis is that it also expands the theoretical guarantee of SSC to handle cases when the subspaces have arbitrarily small canonical angles but are “nearly independent”. We propose low-rank representation (LRR) to segment data drawn from a union of multiple linear (or affine) subspaces. Given a set of data vectors, LRR seeks the lowest-rank representation among all the candidates that represent all vectors as the linear combination of the bases in a dictionary. Unlike the well-known sparse representation (SR), which computes the sparsest representation of each data vector individually, LRR aims at finding the lowest-rank representation of a collection of vectors jointly. LRR better captures the global structure of data, giving a more effective tool for robust subspace segmentation from corrupted data. Both theoretical and experimental results show that LRR is a promising tool for subspace segmentation. Low-rank learning has attracted much attention recently due to its efficacy in a rich variety of real-world tasks, e.g., subspace segmentation and image categorization. Most low-rank methods are incapable of capturing low-dimensional subspace for supervised learning tasks, e.g., classification and regression. This paper aims to learn both the discriminant low-rank representation (LRR) and the robust projecting subspace in a supervised manner. To achieve this goal, we cast the problem into a constrained rank minimization framework by adopting the least squares regularization. Naturally, the data label structure tends to resemble that of the corresponding low-dimensional representation, which is derived from the robust subspace projection of clean data by low-rank learning. Moreover, the low-dimensional representation of original data can be paired with some informative structure by imposing an appropriate constraint, e.g., Laplacian regularizer. Therefore, we propose a novel constrained LRR method. The objective function is formulated as a constrained nuclear norm minimization problem, which can be solved by the inexact augmented Lagrange multiplier algorithm. Extensive experiments on image classification, human pose estimation, and robust face recovery have confirmed the superiority of our method.