Many tensor-based data completion methods aim to solve image and video in-painting problems. But, all methods were only developed for a single dataset. In most of real applications, we can usually obtain more than one dataset to reflect one phenomenon, and all the datasets are mutually related in some sense. Thus one question raised whether such the relationship can improve the performance of data completion or not? In the paper, we proposed a novel and efficient method by exploiting the relationship among datasets for multi-video data completion. Numerical results show that the proposed method significantly improve the performance of video in-painting, particularly in the case of very high missing percentage.
In many different field, multi-data analysis or group analysis was also deeply discussed @cite_0 @cite_2 @cite_4 . But the majority of methods focus on common and distinctive component extraction. In this paper, we do not care latent features in each dataset. Thus we can reduce the complexity of algorithms compared to tensor decomposition based method like @cite_5 @cite_8 . A multi-matrix completion method Convex Collective Matrix Factorization (cCMF) was proposed in @cite_4 . Compare to cCMF, The proposed method in this paper can be considered as an extension of cCMF into tensor problem.
@cite_4 @cite_8 @cite_0 @cite_2 @cite_5
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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]. 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]. CANDECOMP PARAFAC (CP) tensor factorization of incomplete data is a powerful technique for tensor completion through explicitly capturing the multilinear latent factors. The existing CP algorithms require the tensor rank to be manually specified, however, the determination of tensor rank remains a challenging problem especially for CP rank . In addition, existing approaches do not take into account uncertainty information of latent factors, as well as missing entries. To address these issues, we formulate CP factorization using a hierarchical probabilistic model and employ a fully Bayesian treatment by incorporating a sparsity-inducing prior over multiple latent factors and the appropriate hyperpriors over all hyperparameters, resulting in automatic rank determination. To learn the model, we develop an efficient deterministic Bayesian inference algorithm, which scales linearly with data size. Our method is characterized as a tuning parameter-free approach, which can effectively infer underlying multilinear factors with a low-rank constraint, while also providing predictive distributions over missing entries. Extensive simulations on synthetic data illustrate the intrinsic capability of our method to recover the ground-truth of CP rank and prevent the overfitting problem, even when a large amount of entries are missing. Moreover, the results from real-world applications, including image inpainting and facial image synthesis, demonstrate that our method outperforms state-of-the-art approaches for both tensor factorization and tensor completion in terms of predictive performance. In this paper we propose novel methods for compression and recovery of multilinear data under limited sampling. We exploit the recently proposed tensor- Singular Value Decomposition (t-SVD)[1], which is a group theoretic framework for tensor decomposition. In contrast to popular existing tensor decomposition techniques such as higher-order SVD (HOSVD), t-SVD has optimality properties similar to the truncated SVD for matrices. Based on t-SVD, we first construct novel tensor-rank like measures to characterize informational and structural complexity of multilinear data. Following that we outline a complexity penalized algorithm for tensor completion from missing entries. As an application, 3-D and 4-D (color) video data compression and recovery are considered. We show that videos with linear camera motion can be represented more efficiently using t-SVD compared to traditional approaches based on vectorizing or flattening of the tensors. Application of the proposed tensor completion algorithm for video recovery from missing entries is shown to yield a superior performance over existing methods. In conclusion we point out several research directions and implications to online prediction of multilinear data. The Canonical Polyadic Decomposition (CPD) of tensors is a powerful tool for analyzing multi-way data and is used extensively to analyze very large and extremely sparse datasets. The bottleneck of computing the CPD is multiplying a sparse tensor by several dense matrices. Algorithms for tensor-matrix products fall into two classes. The first class saves floating point operations by storing a compressed tensor for each dimension of the data. These methods are fast but suffer high memory costs. The second class uses a single uncompressed tensor at the cost of additional floating point operations. In this work, we bridge the gap between the two approaches and introduce the compressed sparse fiber (CSF) a data structure for sparse tensors along with a novel parallel algorithm for tensor-matrix multiplication. CSF offers similar operation reductions as existing compressed methods while using only a single tensor structure. We validate our contributions with experiments comparing against state-of-the-art methods on a diverse set of datasets. Our work uses 58 less memory than the state-of-the-art while achieving 81 of the parallel performance on 16 threads. Tensor (multiway array) factorization and decomposition has become an important tool for data mining. Fueled by the computational power of modern computer researchers can now analyze large-scale tensorial structured data that only a few years ago would have been impossible. Tensor factorizations have several advantages over two-way matrix factorizations including uniqueness of the optimal solution and component identification even when most of the data is missing. Furthermore, multiway decomposition techniques explicitly exploit the multiway structure that is lost when collapsing some of the modes of the tensor in order to analyze the data by regular matrix factorization approaches. Multiway decomposition is being applied to new fields every year and there is no doubt that the future will bring many exciting new applications. The aim of this overview is to introduce the basic concepts of tensor decompositions and demonstrate some of the many benefits and challenges of modeling data multiway for a wide variety of data and problem domains. © 2011 John Wiley & Sons, Inc. WIREs Data Mining Knowl Discov 2011 1 24-40 DOI: 10.1002 widm.1 Many real-world problems, such as web image analysis, document categorization and product recommendation, often exhibit dual-heterogeneity: heterogeneous features obtained in multiple views, and multiple tasks might be related to each other through one or more shared views. To address these Multi-Task Multi-View (MTMV) problems, we propose a tensor-based framework for learning the predictive multilinear structure from the full-order feature interactions within the heterogeneous data. The usage of tensor structure is to strengthen and capture the complex relationships between multiple tasks with multiple views. We further develop efficient multilinear factorization machines (MFMs) that can learn the task-specific feature map and the task-view shared multilinear structures, without physically building the tensor. In the proposed method, a joint factorization is applied to the full-order interactions such that the consensus representation can be learned. In this manner, it can deal with the partially incomplete data without difficulty as the learning procedure does not simply rely on any particular view. Furthermore, the complexity of MFMs is linear in the number of parameters, which makes MFMs suitable to large-scale real-world problems. Extensive experiments on four real-world datasets demonstrate that the proposed method significantly outperforms several state-of-the-art methods in a wide variety of MTMV problems. A novel regularizer of the PARAFAC decomposition factors capturing the tensor's rank is proposed in this paper, as the key enabler for completion of three-way data arrays with missing entries. Set in a Bayesian framework, the tensor completion method incorporates prior information to enhance its smoothing and prediction capabilities. This probabilistic approach can naturally accommodate general models for the data distribution, lending itself to various fitting criteria that yield optimum estimates in the maximum-a-posteriori sense. In particular, two algorithms are devised for Gaussian- and Poisson-distributed data, that minimize the rank-regularized least-squares error and Kullback-Leibler divergence, respectively. The proposed technique is able to recover the “ground-truth” tensor rank when tested on synthetic data, and to complete brain imaging and yeast gene expression datasets with 50 and 15 of missing entries respectively, resulting in recovery errors at -11 dB and -15 dB. In this paper, two supervised methods for enhancing the classification accuracy of the Nonnegative Matrix Factorization (NMF) algorithm are presented. The idea is to extend the NMF algorithm in order to extract features that enforce not only the spatial locality, but also the separability between classes in a discriminant manner. The first method employs discriminant analysis in the features derived from NMF. In this way, a two-phase discriminant feature extraction procedure is implemented, namely NMF plus Linear Discriminant Analysis (LDA). The second method incorporates the discriminant constraints inside the NMF decomposition. Thus, a decomposition of a face to its discriminant parts is obtained and new update rules for both the weights and the basis images are derived. The introduced methods have been applied to the problem of frontal face verification using the well-known XM2VTS database. Both methods greatly enhance the performance of NMF for frontal face verification Tensor CANDECOMP PARAFAC (CP) decomposition has wide applications in statistical learning of latent variable models and in data mining. In this paper, we propose fast and randomized tensor CP decomposition algorithms based on sketching. We build on the idea of count sketches, but introduce many novel ideas which are unique to tensors. We develop novel methods for randomized computation of tensor contractions via FFTs, without explicitly forming the tensors. Such tensor contractions are encountered in decomposition methods such as tensor power iterations and alternating least squares. We also design novel colliding hashes for symmetric tensors to further save time in computing the sketches. We then combine these sketching ideas with existing whitening and tensor power iterative techniques to obtain the fastest algorithm on both sparse and dense tensors. The quality of approximation under our method does not depend on properties such as sparsity, uniformity of elements, etc. We apply the method for topic modeling and obtain competitive results. We propose a method for unsupervised linear feature extraction through tensor decomposition. The linear feature extraction can be formulated as a canonical polyadic decomposition (CPD) of a third-order tensor when transformation matrix is constrained to be equal to the Khatri-Rao product of two matrices. Therefore, standard algorithms for computing CPD decomposition can be used for feature extraction. The proposed method is validated on publicly available low-resolutionmass spectra of cancerous and non-cancerous samples. Obtained results indicate that this approach could be of practical importance in analysis of protein expression profiles. 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. Tensor factorization has become a popular method for learning from multi-relational data. In this context, the rank of the factorization is an important parameter that determines runtime as well as generalization ability. To identify conditions under which factorization is an efficient approach for learning from relational data, we derive upper and lower bounds on the rank required to recover adjacency tensors. Based on our findings, we propose a novel additive tensor factorization model to learn from latent and observable patterns on multi-relational data and present a scalable algorithm for computing the factorization. We show experimentally both that the proposed additive model does improve the predictive performance over pure latent variable methods and that it also reduces the required rank — and therefore runtime and memory complexity — significantly. We introduce an online tensor decomposition based approach for two latent variable modeling problems namely, (1) community detection, in which we learn the latent communities that the social actors in social networks belong to, and (2) topic modeling, in which we infer hidden topics of text articles. We consider decomposition of moment tensors using stochastic gradient descent. We conduct optimization of multilinear operations in SGD and avoid directly forming the tensors, to save computational and storage costs. We present optimized algorithm in two platforms. Our GPU-based implementation exploits the parallelism of SIMD architectures to allow for maximum speed-up by a careful optimization of storage and data transfer, whereas our CPU-based implementation uses efficient sparse matrix computations and is suitable for large sparse data sets. For the community detection problem, we demonstrate accuracy and computational efficiency on Facebook, Yelp and DBLP data sets, and for the topic modeling problem, we also demonstrate good performance on the New York Times data set. We compare our results to the state-of-the-art algorithms such as the variational method, and report a gain of accuracy and a gain of several orders of magnitude in the execution time. Real-world data are often acquired as a collection of matrices rather than as a single matrix. Such multiblock data are naturally linked and typically share some common features while at the same time exhibiting their own individual features, reflecting the underlying data generation mechanisms. To exploit the linked nature of data, we propose a new framework for common and individual feature extraction (CIFE) which identifies and separates the common and individual features from the multiblock data. Two efficient algorithms termed common orthogonal basis extraction (COBE) are proposed to extract common basis is shared by all data, independent on whether the number of common components is known beforehand. Feature extraction is then performed on the common and individual subspaces separately, by incorporating dimensionality reduction and blind source separation techniques. Comprehensive experimental results on both the synthetic and real-world data demonstrate significant advantages of the proposed CIFE method in comparison with the state-of-the-art. With advances in data collection technologies, multiple data sources are assuming increasing prominence in many applications. Clustering from multiple data sources has emerged as a topic of critical significance in the data mining and machine learning community. Different data sources provide different levels of necessarily detailed knowledge. Thus, combining multiple data sources is pivotal to facilitate the clustering process. However, in reality, the data usually exhibits heterogeneity and incompleteness. The key challenge is how to effectively integrate information from multiple heterogeneous sources in the presence of missing data. Conventional methods mainly focus on clustering heterogeneous data with full information in all sources or at least one source without missing values. In this paper, we propose a more general framework T-MIC (Tensor based Multi-source Incomplete data Clustering) to integrate multiple incomplete data sources. Specifically, we first use the kernel matrices to form an initial tensor across all the multiple sources. Then we formulate a joint tensor factorization process with the sparsity constraint and use it to iteratively push the initial tensor towards a quality-driven exploration of the latent factors by taking into account missing data uncertainty. Finally, these factors serve as features to clustering. Extensive experiments on both synthetic and real datasets demonstrate that our proposed approach can effectively boost clustering performance, even with large amounts of missing data. We introduce an algorithm for a non-negative 3D tensor factorization for the purpose of establishing a local parts feature decomposition from an object class of images. In the past, such a decomposition was obtained using non-negative matrix factorization (NMF) where images were vectorized before being factored by NMF. A tensor factorization (NTF) on the other hand preserves the 2D representations of images and provides a unique factorization (unlike NMF which is not unique). The resulting "factors" from the NTF factorization are both sparse (like with NMF) but also separable allowing efficient convolution with the test image. Results show a superior decomposition to what an NMF can provide on all fronts - degree of sparsity, lack of ghost residue due to invariant parts and efficiency of coding of around an order of magnitude better. Experiments on using the local parts decomposition for face detection using SVM and Adaboost classifiers demonstrate that the recovered features are discriminatory and highly effective for classification. Many problems in computational neuroscience, neuroinformatics, pattern image recognition, signal processing and machine learning generate massive amounts of multidimensional data with multiple aspects and high dimensionality. Tensors (i.e., multi-way arrays) provide often a natural and compact representation for such massive multidimensional data via suitable low-rank approximations. Big data analytics require novel technologies to efficiently process huge datasets within tolerable elapsed times. Such a new emerging technology for multidimensional big data is a multiway analysis via tensor networks (TNs) and tensor decompositions (TDs) which represent tensors by sets of factor (component) matrices and lower-order (core) tensors. Dynamic tensor analysis allows us to discover meaningful hidden structures of complex data and to perform generalizations by capturing multi-linear and multi-aspect relationships. We will discuss some fundamental TN models, their mathematical and graphical descriptions and associated learning algorithms for large-scale TDs and TNs, with many potential applications including: Anomaly detection, feature extraction, classification, cluster analysis, data fusion and integration, pattern recognition, predictive modeling, regression, time series analysis and multiway component analysis. Keywords: Large-scale HOSVD, Tensor decompositions, CPD, Tucker models, Hierarchical Tucker (HT) decomposition, low-rank tensor approximations (LRA), Tensorization Quantization, tensor train (TT QTT) - Matrix Product States (MPS), Matrix Product Operator (MPO), DMRG, Strong Kronecker Product (SKP). This paper presents a detailed empirical study of 12 generative approaches to text clustering, obtained by applying four types of document-to-cluster assignment strategies (hard, stochastic, soft and deterministic annealing (DA) based assignments) to each of three base models, namely mixtures of multivariate Bernoulli, multinomial, and von Mises-Fisher (vMF) distributions. A large variety of text collections, both with and without feature selection, are used for the study, which yields several insights, including (a) showing situations wherein the vMF-centric approaches, which are based on directional statistics, fare better than multinomial model-based methods, and (b) quantifying the trade-off between increased performance of the soft and DA assignments and their increased computational demands. We also compare all the model-based algorithms with two state-of-the-art discriminative approaches to document clustering based, respectively, on graph partitioning (CLUTO) and a spectral coclustering method. Overall, DA and CLUTO perform the best but are also the most computationally expensive. The vMF models provide good performance at low cost while the spectral coclustering algorithm fares worse than vMF-based methods for a majority of the datasets. This paper presents the optimized design and implementation of sparse tensor-times-dense matrix multiply (SpTTM) for CPU and GPU platforms. This primitive is a critical bottleneck in data analysis and mining applications based on tensor methods, such as the Tucker decomposition. We first design and implement sequential SpTTM to avoid explicit data transformations between a tensor and a matrix, which is the conventional approach. We further optimize SpTTM on multicore CPU and GPU systems by parallelizing, avoiding locks, and exploiting data locality. Our sequential SpTTM is up to 3.5× faster than the SpTTM from Tensor Toolbox and 1.5× over that from Cyclops Tensor Framework. Our parallel algorithms show 4.1× speedup on multicore Intel Core i7 and 18.8× speedup on NVIDIA K40c GPU over our sequential SpTTM respectively.