This survey provides an overview of higher-order tensor decompositions, their applications, and available software. A tensor is a multidimensional or N -way array. Decompositions of higher-order tensors (i.e., N -way arrays with N ≥ 3) have applications in psychometrics, chemometrics, signal processing, numerical linear algebra, computer vision, numerical analysis, data mining, neuroscience, graph analysis, etc. Two particular tensor decompositions can be considered to be higher-order extensions of the matrix singular value decompo- sition: CANDECOMP/PARAFAC (CP) decomposes a tensor as a sum of rank-one tensors, and the Tucker decomposition is a higher-order form of principal components analysis. There are many other tensor decompositions, including INDSCAL, PARAFAC2, CANDELINC, DEDICOM, and PARATUCK2 as well as nonnegative variants of all of the above. The N-way Toolbox and Tensor Toolbox, both for MATLAB, and the Multilinear Engine are examples of software packages for working with tensors.

Multiway data analysis captures multilinear structures in higher-order datasets, where data have more than two modes. Standard two-way methods commonly applied on matrices often fail to find the underlying structures in multiway arrays. With increasing number of application areas, multiway data analysis has become popular as an exploratory analysis tool. We provide a review of significant contributions in literature on multiway models, algorithms as well as their applications in diverse disciplines including chemometrics, neuroscience, computer vision, and social network analysis. 1.

Many data analysis applications deal with large matrices and involve approximating the matrix using a small number of “components.” Typically, these components are linear combinations of the rows and columns of the matrix, and are thus difficult to interpret in terms of the original features of the input data. In this paper, we propose and study matrix approximations that are explicitly expressed in terms of a small number of columns and/or rows of the data matrix, and thereby more amenable to interpretation in terms of the original data. Our main algorithmic results are two randomized algorithms which take as input an m × n matrix A and a rank parameter k. In our first algorithm, C is chosen, and we let A ′ = CC + A, where C + is the Moore–Penrose generalized inverse of C. In our second algorithm C, U, R are chosen, and we let A ′ = CUR. (C and R are matrices that consist of actual columns and rows, respectively, of A, and U is a generalized inverse of their intersection.) For each algorithm, we show that with probability at least 1 − δ, ‖A − A ′ ‖F ≤ (1 + ɛ) ‖A − Ak‖F, where Ak is the “best ” rank-k approximation provided by truncating the SVD of A, and where ‖X‖F is the Frobenius norm of the matrix X. The number of columns of C and rows of R is a low-degree polynomial in k, 1/ɛ, and log(1/δ). Both the Numerical Linear Algebra community and the Theoretical Computer Science community have studied variants

We consider the problem of selecting the “best ” subset of exactly k columns from an m × n matrix A. In particular, we present and analyze a novel two-stage algorithm that runs in O(min{mn 2, m 2 n}) time and returns as output an m × k matrix C consisting of exactly k columns of A. In the first stage (the randomized stage), the algorithm randomly selects O(k log k) columns according to a judiciously-chosen probability distribution that depends on information in the topk right singular subspace of A. In the second stage (the deterministic stage), the algorithm applies a deterministic column-selection procedure to select and return exactly k columns from the set of columns selected in the first stage. Let C be the m × k matrix containing those k columns, let PC denote the projection matrix onto the span of those columns, and let Ak denote the “best ” rank-k approximation to the matrix A as computed with the singular value decomposition. Then, we prove that ‖A − PCA‖2 ≤ O k 3 4 log 1

Principal Components Analysis (PCA) is the predominant linear dimensionality reduction technique, and has been widely applied on datasets in all scientific domains. We consider, both theoretically and empirically, the topic of unsupervised feature selection for PCA, by leveraging algorithms for the so-called Column Subset Selection Problem (CSSP). In words, the CSSP seeks the“best”subset of exactly k columns from an m×n data matrix A, and has been extensively studied in the Numerical Linear Algebra community. We present a novel two-stage algorithm for the CSSP. From a theoretical perspective, for small to moderate values of k, this algorithm significantly improves upon the best previously-existing results [24, 12] for the CSSP. From an empirical perspective, we evaluate this algorithm as an unsupervised feature selection strategy in three application domains of modern statistical data analysis: finance, document-term data, and genetics. We pay particular attention to how this algorithm may be used to select representative or landmark features from an object-feature matrix in an unsupervised manner. In all three application domains, we are able to identify k landmark features, i.e., columns of the data matrix, that capture nearly the same amount of information as does the subspace that is spanned by the top k “eigenfeatures.”

Abstract—Social Tagging is the process by which many users add metadata in the form of keywords, to annotate and categorize items (songs, pictures, web links, products, etc.). Social tagging systems (STSs) can provide three different types of recommendations: They can recommend 1) tags to users, based on what tags other users have used for the same items, 2) items to users, based on tags they have in common with other similar users, and 3) users with common social interest, based on common tags on similar items. However, users may have different interests for an item, and items may have multiple facets. In contrast to the current recommendation algorithms, our approach develops a unified framework to model the three types of entities that exist in a social tagging system: users, items, and tags. These data are modeled by a 3-order tensor, on which multiway latent semantic analysis and dimensionality reduction is performed using both the Higher Order Singular Value Decomposition (HOSVD) method and the Kernel-SVD smoothing technique. We perform experimental comparison of the proposed method against state-of-the-art recommendation algorithms with two real data sets (Last.fm and BibSonomy). Our results show significant improvements in terms of effectiveness measured through recall/precision. Index Terms—Social tags, recommender systems, tensors, HOSVD. Ç

Tensors naturally model many real world processes which generate multi-aspect data. Such processes appear in many different research disciplines, e.g, chemometrics, computer vision, psychometrics and neuroimaging analysis. Tensor decompositions such as the Tucker decomposition are used to analyze multi-aspect data and extract latent factors, which capture the multilinear data structure. Such decompositions are powerful mining tools for extracting patterns from large data volumes. However, most frequently used algorithms for such decompositions involve the computationally expensive Singular Value Decomposition. In this paper we propose MACH, a new sampling algorithm to compute such decompositions. Our method is of significant practical value for tensor streams, such as environmental monitoring systems, IP traffic matrices over time, where large amounts of data are accumulated and the analysis is computationally intensive but also in “post-mortem ” data analysis cases where the tensor does not fit in the available memory. We provide the theoretical analysis of our proposed method and verify its efficacy on synthetic data and two real world monitoring system applications. Categories and Subject Descriptors:

Abstract. Dynamic textures are sequences of images showing temporal regularity, such as smoke, flames, flowing water, or moving grass. Despite being a multidimensional signal, existing models reshape the dynamic texture into a 2D signal for analysis. In this article, we propose to directly decompose the multidimensional (tensor) signal, free from reshaping operations. We show that decomposition techniques originally applied to study psychometric or chemometric data can be used for this purpose. Since spatial, time, and color information are analyzed at the same time, such techniques permit to obtain more compact models. Only one third or less model coefficients are needed for the same quality and synthesis cost of 2D based models, as illustrated by experiments on real dynamic textures. 1

Low-rank approximations which are computed from selected rows and columns of a given data matrix have attracted considerable attention lately. They have been proposed as an alternative to the SVD because they naturally lead to interpretable decompositions which was shown to be successful in application such as fraud detection, fMRI segmentation, and collaborative filtering. The CUR decomposition of large matrices, for example, samples rows and columns according to a probability distribution that depends on the Euclidean norm of rows or columns or on other measures of statistical leverage. At the same time, there are various deterministic approaches that do not resort to sampling and were found to often yield factorization of superior quality with respect to reconstruction accuracy. However, these are hardly applicable to large matrices as they typically suffer from high computational costs. Consequently, many practitioners in the field of data mining have abandon deterministic approaches in favor of randomized ones when dealing with today’s large-scale data sets. In this paper, we empirically disprove this prejudice. We do so by introducing a novel, linear-time, deterministic CUR approach that adopts the recently introduced Simplex Volume Maximization approach for column selection. The latter has already been proven to be successful for NMF-like decompositions of matrices of billions of entries. Our exhaustive empirical study on more than 30 synthetic and real-world data sets demonstrates that it is also beneficial for CUR-like decompositions. Compared to other deterministic CUR-like methods, it provides comparable reconstruction quality but operates much faster so that it easily scales to matrices of billions of elements. Compared to sampling-based methods, it provides competitive reconstruction quality while staying in the same run-time complexity class. 1

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