Sparse subspace clustering (SSC) is one of the current state-of-the-art method for partitioning data points into the union of subspaces, with strong theoretical guarantees. However, it is not practical for large data sets as it requires solving a LASSO problem for each data point, where the number of variables in each LASSO problem is the number of data points. To improve the scalability of SSC, we propose to select a few sets of anchor points using a randomized hierarchical clustering method, and, for each set of anchor points, solve the LASSO problems for each data point allowing only anchor points to have a non-zero weight (this reduces drastically the number of variables). This generates a multilayer graph where each layer corresponds to a different set of anchor points. Using the Grassmann manifold of orthogonal matrices, the shared connectivity among the layers is summarized within a single subspace. Finally, we use @math -means clustering within that subspace to cluster the data points, similarly as done by spectral clustering in SSC. We show on both synthetic and real-world data sets that the proposed method not only allows SSC to scale to large-scale data sets, but that it is also much more robust as it performs significantly better on noisy data and on data with close susbspaces and outliers, while it is not prone to oversegmentation.
Several methods in the literature have already addressed the scalability of SSC. One of the earliest attempts is the method referred to as scalable SSC (SSSC) @cite_20 that applies SSC on a randomly selected subset of data points and assigns the rest of the data points based on the obtained clusters. This approach is very sensitive to the selection method, and suboptimal because the data points which were not selected are not taken into account for generating the clusters.
@cite_20
2040329636 2160915541 2759684359 2766915143
In this paper, we address two problems in Sparse Subspace Clustering algorithm (SSC), i.e., scalability issue and out-of-sample problem. SSC constructs a sparse similarity graph for spectral clustering by using l1-minimization based coefficients, has achieved state-of-the-art results for image clustering and motion segmentation. However, the time complexity of SSC is proportion to the cubic of problem size such that it is inefficient to apply SSC into large scale setting. Moreover, SSC does not handle with out-of-sample data that are not used to construct the similarity graph. For each new datum, SSC needs recalculating the cluster membership of the whole data set, which makes SSC is not competitive in fast online clustering. To address the problems, this paper proposes out-of-sample extension of SSC, named as Scalable Sparse Subspace Clustering (SSSC), which makes SSC feasible to cluster large scale data sets. The solution of SSSC adopts a "sampling, clustering, coding, and classifying" strategy. Extensive experimental results on several popular data sets demonstrate the effectiveness and efficiency of our method comparing with the state-of-the-art algorithms. This paper studies the subspace segmentation problem. Given a set of data points drawn from a union of subspaces, the goal is to partition them into their underlying subspaces they were drawn from. The spectral clustering method is used as the framework. It requires to find an affinity matrix which is close to block diagonal, with nonzero entries corresponding to the data point pairs from the same subspace. In this work, we argue that both sparsity and the grouping effect are important for subspace segmentation. A sparse affinity matrix tends to be block diagonal, with less connections between data points from different subspaces. The grouping effect ensures that the highly corrected data which are usually from the same subspace can be grouped together. Sparse Subspace Clustering (SSC), by using l1-minimization, encourages sparsity for data selection, but it lacks of the grouping effect. On the contrary, Low-Rank Representation (LRR), by rank minimization, and Least Squares Regression (LSR), by l2-regularization, exhibit strong grouping effect, but they are short in subset selection. Thus the obtained affinity matrix is usually very sparse by SSC, yet very dense by LRR and LSR. In this work, we propose the Correlation Adaptive Subspace Segmentation (CASS) method by using trace Lasso. CASS is a data correlation dependent method which simultaneously performs automatic data selection and groups correlated data together. It can be regarded as a method which adaptively balances SSC and LSR. Both theoretical and experimental results show the effectiveness of CASS. In this paper, we present a deep extension of Sparse Subspace Clustering, termed Deep Sparse Subspace Clustering (DSSC). Regularized by the unit sphere distribution assumption for the learned deep features, DSSC can infer a new data affinity matrix by simultaneously satisfying the sparsity principle of SSC and the nonlinearity given by neural networks. One of the appealing advantages brought by DSSC is: when original real-world data do not meet the class-specific linear subspace distribution assumption, DSSC can employ neural networks to make the assumption valid with its hierarchical nonlinear transformations. To the best of our knowledge, this is among the first deep learning based subspace clustering methods. Extensive experiments are conducted on four real-world datasets to show the proposed DSSC is significantly superior to 12 existing methods for subspace clustering. In this paper, we proposed a novel method for effective salient object detection by designing a chained multi-scale fully convolutional network (CMSFCN). CMSFCN contained multiple single-scale fully convolutional networks (SSFCNs), which were integrated successively by using chained connections and generated saliency prediction results from coarse to fine. The chained connections not only combined the saliency prediction result from previous SSFCN with the input image of current SSFCN, but also combined the intermediate features from previous SSFCN and current SSFCN. With these chained connections, the sequential SSFCNs in CMSFCN automatically learned complemental and discriminative features to improve the saliency predictions progressively. Therefore, after jointly training CMSFCN with an end-to-end manner, precise saliency prediction results were produced under a coarse-to-fine behaviour. Compared with seven state-of-the-art CNN based salient object detection approaches over five benchmark datasets, experimental results demonstrated the efficiency and effectiveness of CMSFCN.