In real-world applications, not all instances in multi-view data are fully represented. To deal with incomplete multi-view data, traditional multi-view algorithms usually throw away the incomplete instances, resulting in loss of available information. To overcome this loss, Incomplete Multi-view Learning (IML) has become a hot research topic. In this paper, we propose a general IML framework for unifying existing IML methods and gaining insight into IML. The proposed framework jointly performs embedding learning and low-rank approximation. Concretely, it approximates the incomplete data by a set of low-rank matrices and learns a full and common embedding by linear transformation. Several existing IML methods can be unified as special cases of the framework. More interestingly, some linear transformation based full-view methods can be adapted to IML directly with the guidance of the framework. This bridges the gap between full multi-view learning and IML. Moreover, the framework can provide guidance for developing new algorithms. For illustration, within the framework, we propose a specific method, termed as Incomplete Multi-view Learning with Block Diagonal Representation (IML-BDR). Based on the assumption that the sampled examples have approximate linear subspace structure, IML-BDR uses the block diagonal structure prior to learn the full embedding, which would lead to more correct clustering. A convergent alternating iterative algorithm with the Successive Over-Relaxation (SOR) optimization technique is devised for optimization. Experimental results on various datasets demonstrate the effectiveness of IML-BDR.
PVC @cite_24 is originally designed for two-view data in the missing view setting. It learns a full representation from incomplete multi-view data based on the NMF. Denote @math as the examples presented in both views, and denote @math , @math as the examples only presented in the first view and the second view, respectively. The optimization problems of PVC is From codes (http: lamda.nju.edu.cn code .ashx) published by the authors, it can be known that the @math -norm in PVC @cite_24 is actually the @math -norm of matrix. where @math @math is the latent representation for the @math -th view, and @math is the basis matrix, @math is the dimension of the latent space, and @math is the trade-off parameter for the regularization terms.
@cite_24
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Multiview representations reveal the fundamental attributes of the studied instances from different perspectives. Some common perspectives are reviewed by multiple views simultaneously, while some specific ones are reflected by individual views. That is, there are two kinds of properties embedded in the multiview data: 1) consistency and 2) complementarity. Different from most multiview learning approaches only focusing on either consistency or complementarity, this paper proposes a novel semisupervised multiview learning algorithm, called partially shared latent factor (PSLF) learning, which jointly exploits both consistent and complementary information among multiple views. In PSLF, a nonnegative matrix factorization (NMF)-based formulation is adopted to learn a compact and comprehensive partially shared latent representation, which is composed of common latent factors shared by multiple views and some specific latent factors to each view. With the learned representations of multiview data, we introduce a robust sparse regression model to predict the cluster labels of labeled data. By integrating the NMF-based model and the regression model, we obtain a unified formulation and propose a multiplicative-based alternative algorithm for optimization. In addition, PSLF can learn the weights of different views adaptively according to the reconstruction precisions of data matrices. Our experimental study indicates different multiview data that contains consistent and complementary information in different degrees. In addition, the encouraging results of the proposed algorithm are achieved in comparison with the state-of-the-art algorithms on real-world data sets.