Abstract. 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 psycho-metrics, chemometrics, signal processing, numerical linear algebra, computer vision, nu-merical analysis, data mining, neuroscience, graph analysis, and elsewhere. Two particular tensor decompositions can be considered to be higher-order extensions of the matrix sin-gular value decomposition: CANDECOMP/PARAFAC (CP) decomposes a tensor as a sum of rank-one tensors, and the Tucker decomposition is a higher-order form of principal component analysis. There are many other tensor decompositions, including INDSCAL, PARAFAC2, CANDELINC, DEDICOM, and PARATUCK2 as well as nonnegative vari-ants of all of the above. The N-way Toolbox, Tensor Toolbox, and Multilinear Engine are examples of software packages for working with tensors. Key words. tensor decompositions, multiway arrays, multilinear algebra, parallel factors (PARAFAC), canonical decomposition (CANDECOMP), higher-order principal components analysis (Tucker), higher-order singular value decomposition (HOSVD) AMS subject classifications. 15A69, 65F99 DOI. 10.1137/07070111

Tensors (also known as mutidimensional arrays or N -way arrays) are used in a variety of applications ranging from chemometrics to psychometrics. We describe four MATLAB classes for tensor manipulations that can be used for fast algorithm prototyping. The tensor class extends the functionality of MAT-LAB's multidimensional arrays by supporting additional operations such as tensor multiplication. The tensor as matrix class supports the "matricization" of a tensor, i.e., the conversion of a tensor to a matrix (and vice versa), a commonly used operation in many algorithms. Two additional classes represent tensors stored in decomposed formats: cp tensor and tucker tensor. We descibe all of these classes and then demonstrate their use by showing how to implement several tensor algorithms that have appeared in the literature.

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Linear algebra is a powerful and proven tool in web search. Techniques, such as the PageRank algorithm of Brin and Page and the HITS algorithm of Kleinberg, score web pages based on the principal eigenvector (or singular vector) of a particular non-negative matrix that captures the hyperlink structure of the web graph. We propose and test a new methodology that uses multilinear algebra to elicit more information from a higher-order representation of the hyperlink graph. We start by labeling the edges in our graph with the anchor text of the hyperlinks so that the associated linear algebra representation is a sparse, three-way tensor. The first two dimensions of the tensor represent the web pages while the third dimension adds the anchor text. We then use the rank-1 factors of a multilinear PARAFAC tensor decomposition, which are akin to singular vectors of the SVD, to automatically identify topics in the collection along with the associated authoritative web pages.3 4548

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A standard approach to cross-language information retrieval (CLIR) uses Latent Semantic Analysis (LSA) in conjunction with a multilingual parallel aligned corpus. This approach has been shown to be successful in identifying similar documents across languages -or more precisely, retrieving the most similar document in one language to a query in another language. However, the approach has severe drawbacks when applied to a related task, that of clustering documents 'language-independently', so that documents about similar topics end up closest to one another in the semantic space regardless of their language. The problem is that documents are generally more similar to other documents in the same language than they are to documents in a different language, but on the same topic. As a result, when using multilingual LSA, documents will in practice cluster by language, not by topic.We propose a novel application of PARAFAC2 (which is a variant of PARAFAC, a multi-way generalization of the singular value decomposition [SVD]) to overcome this problem. Instead of forming a single multilingual term-by-document matrix which, under LSA, is subjected to SVD, we form an irregular three-way array, each slice of which is a separate term-by-document matrix for a single language in the parallel corpus. The goal is to compute an SVD for each language such that V (the matrix of right singular vectors) is the same across all languages. Effectively, PARAFAC2 imposes the constraint, not present in standard LSA, that the 'concepts' in all documents in the parallel corpus are the same regardless of language. Intuitively, this constraint makes sense, since the whole purpose of using a parallel corpus is that exactly the same concepts are expressed in the translations.We tested this approach by comparing the performance of PARAFAC2 with standard LSA in solving a particular CLIR problem. From our results, we conclude that PARAFAC2 offers a very promising alternative to LSA not only for multilingual document clustering, but also for solving other problems in cross-language information retrieval.4

DEDICOM is an algebraic model for analyzing intrinsically asymmetric relationships, such as the balance of trade among nations or the flow of information among organizations or individuals. It provides information on latent components in the data that can be regarded as "properties" or "aspects" of the objects, and it finds a few patterns that can be combined to describe many relationships among these components. When we apply this technique to adjacency matrices arising from directed graphs, we obtain a smaller graph that gives an idealized description of its patterns. Three-way DEDICOM is a higher-order extension of the model that has certain uniqueness properties. It allows for a third mode of the data, such as time, and permits the analysis of semantic graphs. We present an improved algorithm for computing three-way DEDICOM on sparse data and demonstrate it by applying it to the adjacency tensor of a semantic graph with time-labeled edges. Our application uses the Enron email corpus, from which we construct a semantic graph corresponding to email exchanges among Enron personnel over a series of 44 months. Meaningful patterns are recovered in which the representation of asymmetries adds insight into the social networks at Enron.