Multi-view clustering is an important and fundamental problem. Many multi-view subspace clustering methods have been proposed and achieved success in real-world applications, most of which assume that all views share a same coefficient matrix. However, the underlying information of multiview data are not exploited effectively under this assumption, since the coefficient matrices of different views should have the same clustering properties rather than be the same among multiple views. To this end, a novel Constrained Bilinear Factorization Multi-view Subspace Clustering (CBF-MSC) method is proposed in this paper. Specifically, the bilinear factorization with an orthonormality constraint and a low-rank constraint is employed for all coefficient matrices to make all coefficient matrices have the same trace-norm instead of being equivalent, so as to explore the consensus information of multi-view data more effectively. Finally, an algorithm based on the Augmented Lagrangian Multiplier (ALM) scheme with alternating direction minimization is designed to optimize the objective function. Comprehensive experiments tested on six benchmark datasets validate the effectiveness and competitiveness of the proposed approach compared with several state-of-the-art approaches.
The matrix factorization is often used for matrix approximation @cite_30 @cite_29 . Our approach takes advantage of the bilinear factorization to construct a coefficient matrix which contains the consensus information of all views. It is noteworthy that our method is different from approaches proposed in @cite_37 @cite_32 @cite_11 . @cite_32 formulates a joint nonnegative matrix factorization to keep the clustering results among multiple views comparable; @cite_11 constructs a deep matrix factorization framework to explore the consensus information by seeking for a common representation of multi-view data. These methods perform the nonnegative matrix factorization @cite_39 on the data matrix, columns of which are data points, directly. However, since statistic properties of different views are diverse, it is risky to employ the matrix factorization on multiple views straightforward to explore the consensus information. As for our proposed CBF-MSC, it employs a constrained bilinear factorization on subspace coefficient matrices, which are self-representations of multiple views with same or similar statistic properties for clustering @cite_9 @cite_16 @cite_6 @cite_21 .
@cite_30 @cite_37 @cite_29 @cite_9 @cite_21 @cite_32 @cite_6 @cite_39 @cite_16 @cite_11
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We propose a novel nonnegative matrix factorization model that aims at finding localized, part-based, representations of nonnegative multivariate data items. Unlike the classical nonnegative matrix factorization (NMF) technique, this new model, denoted "nonsmooth nonnegative matrix factorization" (nsNMF), corresponds to the optimization of an unambiguous cost function designed to explicitly represent sparseness, in the form of nonsmoothness, which is controlled by a single parameter. In general, this method produces a set of basis and encoding vectors that are not only capable of representing the original data, but they also extract highly focalized patterns, which generally lend themselves to improved interpretability. The properties of this new method are illustrated with several data sets. Comparisons to previously published methods show that the new nsNMF method has some advantages in keeping faithfulness to the data in the achieving a high degree of sparseness for both the estimated basis and the encoding vectors and in better interpretability of the factors. Nonnegative Matrix Factorization (NMF) is an effective dimension reduction method for non-negative dyadic data, and has proven to be useful in many areas, such as text mining, bioinformatics and image processing. NMF is usually formulated as a constrained non-convex optimization problem, and many algorithms have been developed for solving it. Recently, a coordinate descent method, called FastHals, has been proposed to solve least squares NMF and is regarded as one of the state-of-the-art techniques for the problem. In this paper, we first show that FastHals has an inefficiency in that it uses a cyclic coordinate descent scheme and thus, performs unneeded descent steps on unimportant variables. We then present a variable selection scheme that uses the gradient of the objective function to arrive at a new coordinate descent method. Our new method is considerably faster in practice and we show that it has theoretical convergence guarantees. Moreover when the solution is sparse, as is often the case in real applications, our new method benefits by selecting important variables to update more often, thus resulting in higher speed. As an example, on a text dataset RCV1, our method is 7 times faster than FastHals, and more than 15 times faster when the sparsity is increased by adding an L1 penalty. We also develop new coordinate descent methods when error in NMF is measured by KL-divergence by applying the Newton method to solve the one-variable sub-problems. Experiments indicate that our algorithm for minimizing the KL-divergence is faster than the Lee & Seung multiplicative rule by a factor of 10 on the CBCL image dataset. Matrix factorization, when the matrix has missing values, has become one of the leading techniques for recommender systems. To handle web-scale datasets with millions of users and billions of ratings, scalability becomes an important issue. Alternating least squares (ALS) and stochastic gradient descent (SGD) are two popular approaches to compute matrix factorization, and there has been a recent flurry of activity to parallelize these algorithms. However, due to the cubic time complexity in the target rank, ALS is not scalable to large-scale datasets. On the other hand, SGD conducts efficient updates but usually suffers from slow convergence that is sensitive to the parameters. Coordinate descent, a classical optimization approach, has been used for many other large-scale problems, but its application to matrix factorization for recommender systems has not been thoroughly explored. In this paper, we show that coordinate descent-based methods have a more efficient update rule compared to ALS and have faster and more stable convergence than SGD. We study different update sequences and propose the CCD++ algorithm, which updates rank-one factors one by one. In addition, CCD++ can be easily parallelized on both multi-core and distributed systems. We empirically show that CCD++ is much faster than ALS and SGD in both settings. As an example, with a synthetic dataset containing 14.6 billion ratings, on a distributed memory cluster with 64 processors, to deliver the desired test RMSE, CCD++ is 49 times faster than SGD and 20 times faster than ALS. When the number of processors is increased to 256, CCD++ takes only 16 s and is still 40 times faster than SGD and 20 times faster than ALS. Recovering low-rank and sparse matrices from incomplete or corrupted observations is an important problem in statistics, machine learning, computer vision, as well as signal and image processing. In theory, this problem can be solved by the natural convex joint mixed relaxations (i.e., l 1 -norm and trace norm) under certain conditions. However, all current provable algorithms suffer from superlinear per-iteration cost, which severely limits their applicability to large-scale problems. In this paper, we propose a scalable, provable and structured robust bilinear factorization (RBF) method to recover low-rank and sparse matrices from missing and grossly corrupted data, i.e., robust matrix completion (RMC), or incomplete and grossly corrupted measurements, i.e., compressive principal component pursuit (CPCP). Specifically, we first present two small-scale matrix trace norm regularized bilinear factorization models for RMC and CPCP problems, in which repetitively calculating SVD of a large-scale matrix is replaced by updating two much smaller factor matrices. Then, we apply the alternating direction method of multipliers (ADMM) to efficiently solve the RMC problems. Finally, we provide the convergence analysis of our algorithm, and extend it to address general CPCP problems. Experimental results verified both the efficiency and effectiveness of our method compared with the state-of-the-art methods.