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
1977580401 2146130798 1572352401 2030927653
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. We use convex relaxation techniques to provide a sequence of regularized low-rank solutions for large-scale matrix completion problems. Using the nuclear norm as a regularizer, we provide a simple and very efficient convex algorithm for minimizing the reconstruction error subject to a bound on the nuclear norm. Our algorithm SOFT-IMPUTE iteratively replaces the missing elements with those obtained from a soft-thresholded SVD. With warm starts this allows us to efficiently compute an entire regularization path of solutions on a grid of values of the regularization parameter. The computationally intensive part of our algorithm is in computing a low-rank SVD of a dense matrix. Exploiting the problem structure, we show that the task can be performed with a complexity of order linear in the matrix dimensions. Our semidefinite-programming algorithm is readily scalable to large matrices; for example SOFT-IMPUTE takes a few hours to compute low-rank approximations of a 106 X 106 incomplete matrix with 107 observed entries, and fits a rank-95 approximation to the full Netflix training set in 3.3 hours. Our methods achieve good training and test errors and exhibit superior timings when compared to other competitive state-of-the-art techniques. Given a large sparse graph, how can we find patterns and anomalies? Several important applications can be modeled as large sparse graphs, e.g., network traffic monitoring, research citation network analysis, social network analysis, and regulatory networks in genes. Low rank decompositions, such as SVD and CUR, are powerful techniques for revealing latent hidden variables and associated patterns from high dimensional data. However, those methods often ignore the sparsity property of the graph, and hence usually incur too high memory and computational cost to be practical. We propose a novel method, the Compact Matrix Decomposition (CMD), to compute sparse low rank approximations. CMD dramatically reduces both the computation cost and the space requirements over existing decomposition methods (SVD, CUR). Using CMD as the key building block, we further propose procedures to efficiently construct and analyze dynamic graphs from real-time application data. We provide theoretical guarantee for our methods, and present results on two real, large datasets, one on network flow data (100GB trace of 22K hosts over one month) and one on DBLP (200MB over 25 years). We show that CMD is often an order of magnitude more efficient than the state of the art (SVD and CUR): it is over 10X faster, but requires less than 1 10 of the space, for the same reconstruction accuracy. Finally, we demonstrate how CMD is used for detecting anomalies and monitoring timeevolving graphs, in which it successfully detects worm-like hierarchical scanning patterns in real network data. In this paper we propose novel methods for completion (from limited samples) and de-noising of multilinear (tensor) data and as an application consider 3-D and 4- D (color) video data completion and de-noising. We exploit the recently proposed tensor-Singular Value Decomposition (t-SVD)[11]. Based on t-SVD, the notion of multilinear rank and a related tensor nuclear norm was proposed in [11] to characterize informational and structural complexity of multilinear data. We first show that videos with linear camera motion can be represented more efficiently using t-SVD compared to the approaches based on vectorizing or flattening of the tensors. Since efficiency in representation implies efficiency in recovery, we outline a tensor nuclear norm penalized algorithm for video completion from missing entries. Application of the proposed algorithm for video recovery from missing entries is shown to yield a superior performance over existing methods. We also consider the problem of tensor robust Principal Component Analysis (PCA) for de-noising 3-D video data from sparse random corruptions. We show superior performance of our method compared to the matrix robust PCA adapted to this setting as proposed in [4].