Learning the "blocking" structure is a central challenge for high dimensional data (e.g., gene expression data). Recently, a sparse singular value decomposition (SVD) has been used as a biclustering tool to achieve this goal. However, this model ignores the structural information between variables (e.g., gene interaction graph). Although typical graph-regularized norm can incorporate such prior graph information to get accurate discovery and better interpretability, it fails to consider the opposite effect of variables with different signs. Motivated by the development of sparse coding and graph-regularized norm, we propose a novel sparse graph-regularized SVD as a powerful biclustering tool for analyzing high-dimensional data. The key of this method is to impose two penalties including a novel graph-regularized norm ( @math ) and @math -norm ( @math ) on singular vectors to induce structural sparsity and enhance interpretability. We design an efficient Alternating Iterative Sparse Projection (AISP) algorithm to solve it. Finally, we apply our method and related ones to simulated and real data to show its efficiency in capturing natural blocking structures.
(2) Graph-regularized SVD. Graph-regularized norm has been used in many different techniques such as nonnegative matrix factorization @cite_26 . However, to our knowledge, there is yet no study to incorporate graph-regularized norm into the sparse SVD framework. We believe that sparse graph-regularized SVD is a promising tool as explored in other problems @cite_10 @cite_3 @cite_13 to enforce the smoothness of variable coefficients. In addition, @math is widely used to enforce the structural sparsity at the inter-group level @cite_15 . We can also consider it as a penalty function of SVD in future studies.
@cite_26 @cite_10 @cite_3 @cite_15 @cite_13
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Matrix factorization techniques have been frequently applied in information retrieval, computer vision, and pattern recognition. Among them, Nonnegative Matrix Factorization (NMF) has received considerable attention due to its psychological and physiological interpretation of naturally occurring data whose representation may be parts based in the human brain. On the other hand, from the geometric perspective, the data is usually sampled from a low-dimensional manifold embedded in a high-dimensional ambient space. One then hopes to find a compact representation,which uncovers the hidden semantics and simultaneously respects the intrinsic geometric structure. In this paper, we propose a novel algorithm, called Graph Regularized Nonnegative Matrix Factorization (GNMF), for this purpose. In GNMF, an affinity graph is constructed to encode the geometrical information and we seek a matrix factorization, which respects the graph structure. Our empirical study shows encouraging results of the proposed algorithm in comparison to the state-of-the-art algorithms on real-world problems. In many applications, the data, such as web pages and research papers, contain relation (link) structure among entities in addition to textual content information. Matrix factorization (MF) methods, such as latent semantic indexing (LSI), have been successfully used to map either content information or relation information into a lower-dimensional latent space for subsequent processing. However, how to simultaneously model both the relation information and the content information effectively with an MF framework is still an open research problem. In this paper, we propose a novel MF method called relation regularized matrix factorization (RRMF) for relational data analysis. By using relation information to regularize the content MF procedure, RRMF seamlessly integrates both the relation information and the content information into a principled framework. We propose a linear-time learning algorithm with convergence guarantee to learn the parameters of RRMF. Extensive experiments on real data sets show that RRMF can achieve state-of-the-art performance. We present an extension of sparse PCA, or sparse dictionary learning, where the sparsity patterns of all dictionary elements are structured and constrained to belong to a prespecified set of shapes. This is based on a structured regularization recently introduced by [1]. While classical sparse priors only deal with , the regularization we use encodes higher-order information about the data. We propose an efficient and simple optimization procedure to solve this problem. Experiments with two practical tasks, face recognition and the study of the dynamics of a protein complex, demonstrate the benefits of the proposed structured approach over unstructured approaches. We propose a new penalty function which, when used as regularization for empirical risk minimization procedures, leads to sparse estimators. The support of the sparse vector is typically a union of potentially overlapping groups of co-variates defined a priori, or a set of covariates which tend to be connected to each other when a graph of covariates is given. We study theoretical properties of the estimator, and illustrate its behavior on simulated and breast cancer gene expression data. Sparse coding has received an increasing amount of interest in recent years. It is an unsupervised learning algorithm, which finds a basis set capturing high-level semantics in the data and learns sparse coordinates in terms of the basis set. Originally applied to modeling the human visual cortex, sparse coding has been shown useful for many applications. However, most of the existing approaches to sparse coding fail to consider the geometrical structure of the data space. In many real applications, the data is more likely to reside on a low-dimensional submanifold embedded in the high-dimensional ambient space. It has been shown that the geometrical information of the data is important for discrimination. In this paper, we propose a graph based algorithm, called graph regularized sparse coding, to learn the sparse representations that explicitly take into account the local manifold structure of the data. By using graph Laplacian as a smooth operator, the obtained sparse representations vary smoothly along the geodesics of the data manifold. The extensive experimental results on image classification and clustering have demonstrated the effectiveness of our proposed algorithm.