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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Low rank models have been widely used for the representation of shape, appearance or motion in computer vision problems. Traditional approaches to fit low rank models make use of an explicit bilinear factorization. These approaches benefit from fast numerical methods for optimization and easy kernelization. However, they suffer from serious local minima problems depending on the loss function and the amount type of missing data. Recently, these low-rank models have alternatively been formulated as convex problems using the nuclear norm regularizer, unlike factorization methods, their numerical solvers are slow and it is unclear how to kernelize them or to impose a rank a priori. This paper proposes a unified approach to bilinear factorization and nuclear norm regularization, that inherits the benefits of both. We analyze the conditions under which these approaches are equivalent. Moreover, based on this analysis, we propose a new optimization algorithm and a "rank continuation'' strategy that outperform state-of-the-art approaches for Robust PCA, Structure from Motion and Photometric Stereo with outliers and missing data. With the increasing availability of annotated multimedia data on the Internet, techniques are in demand that allow for a principled joint processing of different types of data. Multiview learning and multiview clustering attempt to identify latent components in different features spaces in a simultaneous manner. The resulting basis vectors or centroids faithfully represent the different views on the data but are implicitly coupled and they were jointly estimated. This opens new avenues to problems such as label prediction, image retrieval, or semantic grouping. In this paper, we present a new model for multiview clustering that extends traditional non-negative matrix factorization to the joint factorization of different data matrices. Accordingly, the technique provides a new approach to the joint treatment of image parts and attributes. First experiments in image segmentation and multiview clustering of image features and image labels show promising results and indicate that the proposed method offers a common framework for image analysis on different levels of abstraction. In this paper, we explore the problem of multiview subspace clustering. We introduce a low-rank tensor constraint to explore the complementary information from multiple views and, accordingly, establish a novel method called Low-rank Tensor constrained Multiview Subspace Clustering (LT-MSC). Our method regards the subspace representation matrices of different views as a tensor, which captures dexterously the high order correlations underlying multiview data. Then the tensor is equipped with a low-rank constraint, which models elegantly the cross information among different views, reduces effectually the redundancy of the learned subspace representations, and improves the accuracy of clustering as well. The inference process of the affinity matrix for clustering is formulated as a tensor nuclear norm minimization problem, constrained with an additional L2,1-norm regularizer and some linear equalities. The minimization problem is convex and thus can be solved efficiently by an Augmented Lagrangian Alternating Direction Minimization (AL-ADM) method. Extensive experimental results on four benchmark datasets show the effectiveness of our proposed LT-MSC method. Many real-world datasets are comprised of different representations or views which often provide information complementary to each other. To integrate information from multiple views in the unsupervised setting, multiview clustering algorithms have been developed to cluster multiple views simultaneously to derive a solution which uncovers the common latent structure shared by multiple views. In this paper, we propose a novel NMFbased multi-view clustering algorithm by searching for a factorization that gives compatible clustering solutions across multiple views. The key idea is to formulate a joint matrix factorization process with the constraint that pushes clustering solution of each view towards a common consensus instead of fixing it directly. The main challenge is how to keep clustering solutions across different views meaningful and comparable. To tackle this challenge, we design a novel and effective normalization strategy inspired by the connection between NMF and PLSA. Experimental results on synthetic and several real datasets demonstrate the effectiveness of our approach. With advances in information acquisition technologies, multi-view data become ubiquitous. Multi-view learning has thus become more and more popular in machine learning and data mining fields. Multi-view unsupervised or semi-supervised learning, such as co-training, co-regularization has gained considerable attention. Although recently, multi-view clustering (MVC) methods have been developed rapidly, there has not been a survey to summarize and analyze the current progress. Therefore, this paper reviews the common strategies for combining multiple views of data and based on this summary we propose a novel taxonomy of the MVC approaches. We further discuss the relationships between MVC and multi-view representation, ensemble clustering, multi-task clustering, multi-view supervised and semi-supervised learning. Several representative real-world applications are elaborated. To promote future development of MVC, we envision several open problems that may require further investigation and thorough examination. Is perception of the whole based on perception of its parts? There is psychological1 and physiological2,3 evidence for parts-based representations in the brain, and certain computational theories of object recognition rely on such representations4,5. But little is known about how brains or computers might learn the parts of objects. Here we demonstrate an algorithm for non-negative matrix factorization that is able to learn parts of faces and semantic features of text. This is in contrast to other methods, such as principal components analysis and vector quantization, that learn holistic, not parts-based, representations. Non-negative matrix factorization is distinguished from the other methods by its use of non-negativity constraints. These constraints lead to a parts-based representation because they allow only additive, not subtractive, combinations. When non-negative matrix factorization is implemented as a neural network, parts-based representations emerge by virtue of two properties: the firing rates of neurons are never negative and synaptic strengths do not change sign. In this paper, we address the subspace clustering problem. Given a set of data samples (vectors) approximately drawn from a union of multiple subspaces, our goal is to cluster the samples into their respective subspaces and remove possible outliers as well. To this end, we propose a novel objective function named Low-Rank Representation (LRR), which seeks the lowest rank representation among all the candidates that can represent the data samples as linear combinations of the bases in a given dictionary. It is shown that the convex program associated with LRR solves the subspace clustering problem in the following sense: When the data is clean, we prove that LRR exactly recovers the true subspace structures; when the data are contaminated by outliers, we prove that under certain conditions LRR can exactly recover the row space of the original data and detect the outlier as well; for data corrupted by arbitrary sparse errors, LRR can also approximately recover the row space with theoretical guarantees. Since the subspace membership is provably determined by the row space, these further imply that LRR can perform robust subspace clustering and error correction in an efficient and effective way.