Subspace clustering methods based on data self-expression have become very popular for learning from data that lie in a union of low-dimensional linear subspaces. However, the applicability of subspace clustering has been limited because practical visual data in raw form do not necessarily lie in such linear subspaces. On the other hand, while Convolutional Neural Network (ConvNet) has been demonstrated to be a powerful tool for extracting discriminative features from visual data, training such a ConvNet usually requires a large amount of labeled data, which are unavailable in subspace clustering applications. To achieve simultaneous feature learning and subspace clustering, we propose an end-to-end trainable framework, called Self-Supervised Convolutional Subspace Clustering Network (S @math ConvSCN), that combines a ConvNet module (for feature learning), a self-expression module (for subspace clustering) and a spectral clustering module (for self-supervision) into a joint optimization framework. Particularly, we introduce a dual self-supervision that exploits the output of spectral clustering to supervise the training of the feature learning module (via a classification loss) and the self-expression module (via a spectral clustering loss). Our experiments on four benchmark datasets show the effectiveness of the dual self-supervision and demonstrate superior performance of our proposed approach.
On the other hand, the latent feature space has also been constructed via matrix decomposition, , @cite_21 , @cite_17 . In @cite_21 , a linear transform matrix and a low-rank representation are computed simultaneously; in @cite_17 , a linear transform and a sparse representation are optimized jointly. However, the representation power of the learned linear transform is still limited.
@cite_21 @cite_17
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Low-rank and sparse structures have been profoundly studied in matrix completion and compressed sensing. In this paper, we develop "Go Decomposition" (GoDec) to efficiently and robustly estimate the low-rank part L and the sparse part S of a matrix X = L + S + G with noise G. GoDec alternatively assigns the low-rank approximation of X - S to L and the sparse approximation of X - L to S. The algorithm can be significantly accelerated by bilateral random projections (BRP). We also propose GoDec for matrix completion as an important variant. We prove that the objective value ||X - L - S||2F converges to a local minimum, while L and S linearly converge to local optimums. Theoretically, we analyze the influence of L, S and G to the asymptotic convergence speeds in order to discover the robustness of GoDec. Empirical studies suggest the efficiency, robustness and effectiveness of GoDec comparing with representative matrix decomposition and completion tools, e.g., Robust PCA and OptSpace. In this paper, we address compressed sensing of a low-rank matrix posing the inverse problem as an approximation problem with a specified target rank of the solution. A simple search over the target rank then provides the minimum rank solution satisfying a prescribed data approximation bound. We propose an atomic decomposition providing an analogy between parsimonious representations of a sparse vector and a low-rank matrix and extending efficient greedy algorithms from the vector to the matrix case. In particular, we propose an efficient and guaranteed algorithm named atomic decomposition for minimum rank approximation (ADMiRA) that extends Needell and Tropp's compressive sampling matching pursuit (CoSaMP) algorithm from the sparse vector to the low-rank matrix case. The performance guarantee is given in terms of the rank-restricted isometry property (R-RIP) and bounds both the number of iterations and the error in the approximate solution for the general case of noisy measurements and approximately low-rank solution. With a sparse measurement operator as in the matrix completion problem, the computation in ADMiRA is linear in the number of measurements. Numerical experiments for the matrix completion problem show that, although the R-RIP is not satisfied in this case, ADMiRA is a competitive algorithm for matrix completion. Suppose we are given a matrix that is formed by adding an unknown sparse matrix to an unknown low-rank matrix. Our goal is to decompose the given matrix into its sparse and low-rank components. Such a problem arises in a number of applications in model and system identification and is intractable to solve in general. In this paper we consider a convex optimization formulation to splitting the specified matrix into its components by minimizing a linear combination of the l1 norm and the nuclear norm of the components. We develop a notion of rank-sparsity incoherence, expressed as an uncertainty principle between the sparsity pat- tern of a matrix and its row and column spaces, and we use it to characterize both fundamental identifiability as well as (deterministic) sufficient conditions for exact recovery. Our analysis is geometric in nature with the tangent spaces to the algebraic varieties of sparse and low-rank matrices playing a prominent role. When the sparse and low-rank matrices are drawn from certain natural random ensembles, we show that the sufficient conditions for exact recovery are satisfied with high probability. We conclude with simulation results on synthetic matrix decomposition problems. We introduce an online tensor decomposition based approach for two latent variable modeling problems namely, (1) community detection, in which we learn the latent communities that the social actors in social networks belong to, and (2) topic modeling, in which we infer hidden topics of text articles. We consider decomposition of moment tensors using stochastic gradient descent. We conduct optimization of multilinear operations in SGD and avoid directly forming the tensors, to save computational and storage costs. We present optimized algorithm in two platforms. Our GPU-based implementation exploits the parallelism of SIMD architectures to allow for maximum speed-up by a careful optimization of storage and data transfer, whereas our CPU-based implementation uses efficient sparse matrix computations and is suitable for large sparse data sets. For the community detection problem, we demonstrate accuracy and computational efficiency on Facebook, Yelp and DBLP data sets, and for the topic modeling problem, we also demonstrate good performance on the New York Times data set. We compare our results to the state-of-the-art algorithms such as the variational method, and report a gain of accuracy and a gain of several orders of magnitude in the execution time.