Subspace clustering algorithms are notorious for their scalability issues because building and processing large affinity matrices are demanding. In this paper, we introduce a method that simultaneously learns an embedding space along subspaces within it to minimize a notion of reconstruction error, thus addressing the problem of subspace clustering in an end-to-end learning paradigm. To achieve our goal, we propose a scheme to update subspaces within a deep neural network. This in turn frees us from the need of having an affinity matrix to perform clustering. Unlike previous attempts, our method can easily scale up to large datasets, making it unique in the context of unsupervised learning with deep architectures. Our experiments show that our method significantly improves the clustering accuracy while enjoying cheaper memory footprints.
The SSC by Orthogonal Matching Pursuit (SSC-OMP) @cite_8 is probably the only subspace clustering which could be considered as scalable''. The main idea is to replace the large scale convex optimization procedure with the OMP algorithm in constructing the affinity matrix. Having said this, SSC-OMP makes use of spectral clustering and hence still fails to really push subspace clustering for large scale datasets.
@cite_8
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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 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. Subspace clustering algorithms are notorious for their scalability issues because building and processing large affinity matrices are demanding. In this paper, we introduce a method that simultaneously learns an embedding space along subspaces within it to minimize a notion of reconstruction error, thus addressing the problem of subspace clustering in an end-to-end learning paradigm. To achieve our goal, we propose a scheme to update subspaces within a deep neural network. This in turn frees us from the need of having an affinity matrix to perform clustering. Unlike previous attempts, our method can easily scale up to large datasets, making it unique in the context of unsupervised learning with deep architectures. Our experiments show that our method significantly improves the clustering accuracy while enjoying cheaper memory footprints. Orthogonal matching pursuit (OMP) is a greedy search algorithm popularly being used for the recovery of compressive sensed sparse signals. In this correspondence, we show that if the isometry constant δK+1 of the sensing matrix Φ satisfies δK+1 <; 1 (1 √K+1) then the OMP algorithm can perfectly recover K-sparse signals from the compressed measurements y=Φx. Our bound offers a substantial improvement over the recent result of Davenport and Wakin and also closes gap between the recovery bound and fundamental limit over which the perfect recovery of the OMP cannot be guaranteed.