Subspace clustering is a problem of exploring the low-dimensional subspaces of high-dimensional data. State-of-the-art approaches are designed by following the model of spectral clustering-based method. These methods pay much attention to learn the representation matrix to construct a suitable similarity matrix and overlook the influence of the noise term on subspace clustering. However, the real data are always contaminated by the noise and the noise usually has a complicated statistical distribution. To alleviate this problem, in this paper, we propose a subspace clustering method based on Cauchy loss function (CLF). Particularly, it uses CLF to penalize the noise term for suppressing the large noise mixed in the real data. This is due to that the CLF’s influence function has an upper bound that can alleviate the influence of a single sample, especially the sample with a large noise, on estimating the residuals. Furthermore, we theoretically prove the grouping effect of our proposed method, which means that highly correlated data can be grouped together. Finally, experimental results on five real data sets reveal that our proposed method outperforms several representative clustering methods.
This paper considers the problem of subspace clustering under noise. Specifically, we study the behavior of Sparse Subspace Clustering (SSC) when either adversarial or random noise is added to the unlabeled input data points, which are assumed to be in a union of low-dimensional subspaces. We show that a modified version of SSC is provably effective in correctly identifying the underlying subspaces, even with noisy data. This extends theoretical guarantee of this algorithm to more practical settings and provides justification to the success of SSC in a class of real applications. We propose a method based on sparse representation (SR) to cluster data drawn from multiple low-dimensional linear or affine subspaces embedded in a high-dimensional space. Our method is based on the fact that each point in a union of subspaces has a SR with respect to a dictionary formed by all other data points. In general, finding such a SR is NP hard. Our key contribution is to show that, under mild assumptions, the SR can be obtained exactly' by using l1 optimization. The segmentation of the data is obtained by applying spectral clustering to a similarity matrix built from this SR. Our method can handle noise, outliers as well as missing data. We apply our subspace clustering algorithm to the problem of segmenting multiple motions in video. Experiments on 167 video sequences show that our approach significantly outperforms state-of-the-art methods.