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.
(3) The relationship between SVD and PCA. As we all know, principal component analysis (PCA) can be efficiently solved by using SVD. However, the identified non-sparse principal components can sometimes be difficult to interpret. To solve it, recent studies have developed several different sparse PCA models @cite_16 @cite_6 @cite_12 @cite_11 . However, it is difficult to develop effective algorithms for solving these sparse PCA models. There are a kind of commonly used methods based on regularized SVD for solving them @cite_12 . Moreover, Witten proposed another model based on regularized SVD to ensure the orthogonally of their left singular vectors.
@cite_16 @cite_12 @cite_6 @cite_11
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In sparse principal component analysis we are given noisy observations of a low-rank matrix of dimension @math and seek to reconstruct it under additional sparsity assumptions. In particular, we assume here each of the principal components @math has at most @math non-zero entries. We are particularly interested in the high dimensional regime wherein @math is comparable to, or even much larger than @math . In an influential paper, johnstone2004sparse introduced a simple algorithm that estimates the support of the principal vectors @math by the largest entries in the diagonal of the empirical covariance. This method can be shown to identify the correct support with high probability if @math , and to fail with high probability if @math for two constants @math . Despite a considerable amount of work over the last ten years, no practical algorithm exists with provably better support recovery guarantees. Here we analyze a covariance thresholding algorithm that was recently proposed by KrauthgamerSPCA . On the basis of numerical simulations (for the rank-one case), these authors conjectured that covariance thresholding correctly recover the support with high probability for @math (assuming @math of the same order as @math ). We prove this conjecture, and in fact establish a more general guarantee including higher-rank as well as @math much smaller than @math . Recent lower bounds berthet2013computational, ma2015sum suggest that no polynomial time algorithm can do significantly better. The key technical component of our analysis develops new bounds on the norm of kernel random matrices, in regimes that were not considered before. Principal component analysis (PCA) is a widely used tool for data analysis and dimension reduction in applications throughout science and engineering. However, the principal components (PCs) can sometimes be difficult to interpret, because they are linear combinations of all the original variables. To facilitate interpretation, sparse PCA produces modified PCs with sparse loadings, i.e. loadings with very few non-zero elements. In this paper, we propose a new sparse PCA method, namely sparse PCA via regularized SVD (sPCA-rSVD). We use the connection of PCA with singular value decomposition (SVD) of the data matrix and extract the PCs through solving a low rank matrix approximation problem. Regularization penalties are introduced to the corresponding minimization problem to promote sparsity in PC loadings. An efficient iterative algorithm is proposed for computation. Two tuning parameter selection methods are discussed. Some theoretical results are established to justify the use of sPCA-rSVD when only the data covariance matrix is available. In addition, we give a modified definition of variance explained by the sparse PCs. The sPCA-rSVD provides a uniform treatment of both classical multivariate data and high-dimension-low-sample-size (HDLSS) data. Further understanding of sPCA-rSVD and some existing alternatives is gained through simulation studies and real data examples, which suggests that sPCA-rSVD provides competitive results. In this paper, we study the estimation of the k-dimensional sparse principal sub-space of covariance matrix Σ in the high-dimensional setting. We aim to recover the oracle principal subspace solution, i.e., the principal subspace estimator obtained assuming the true support is known a priori. To this end, we propose a family of estimators based on the semidefinite relaxation of sparse PCA with novel regularizations. In particular, under a weak assumption on the magnitude of the population projection matrix, one estimator within this family exactly recovers the true support with high probability, has exact rank-k, and attains a √s n statistical rate of convergence with s being the subspace sparsity level and n the sample size. Compared to existing support recovery results for sparse PCA, our approach does not hinge on the spiked covariance model or the limited correlation condition. As a complement to the first estimator that enjoys the oracle property, we prove that, another estimator within the family achieves a sharper statistical rate of convergence than the standard semidefinite relaxation of sparse PCA, even when the previous assumption on the magnitude of the projection matrix is violated. We validate the theoretical results by numerical experiments on synthetic datasets. SUMMARY We present a penalized matrix decomposition (PMD), a new framework for computing a rank-K approximation for a matrix. We approximate the matrix X as ˆ X = � K=1 dkukv T , where dk, uk, and