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.

How do we find patterns in author-keyword associations, evolving over time? Or in DataCubes, with product-branchcustomer sales information? Matrix decompositions, like principal component analysis (PCA) and variants, are invaluable tools for mining, dimensionality reduction, feature selection, rule identification in numerous settings like streaming data, text, graphs, social networks and many more. However, they have only two orders, like author and keyword, in the above example. We propose to envision such higher order data as tensors, and tap the vast literature on the topic. However, these methods do not necessarily scale up, let alone operate on semi-infinite streams. Thus, we introduce the dynamic tensor analysis (DTA) method, and its variants. DTA provides a compact summary for high-order and high-dimensional data, and it also reveals the hidden correlations. Algorithmically, we designed DTA very carefully so that it is (a) scalable, (b) space efficient (it does not need to store the past) and (c) fully automatic with no need for user defined parameters. Moreover, we propose STA, a streaming tensor analysis method, which provides a fast, streaming approximation to DTA. We implemented all our methods, and applied them in two real settings, namely, anomaly detection and multi-way latent semantic indexing. We used two real, large datasets, one on network flow data (100GB over 1 month) and one from DBLP (200MB over 25 years). Our experiments show that our methods are fast, accurate and that they find interesting patterns and outliers on the real datasets. 1.

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.

We propose two new multilinear operators for expressing the matrix compositions that are needed in the Tucker and PARAFAC (CANDECOMP) decompositions. The first operator, which we call the Tucker operator, is shorthand for performing an n-mode matrix multiplication for every mode of a given tensor and can be employed to consisely express the Tucker decomposition. The second operator, which we call the Kruskal operator, is shorthand for the sum of the outer-products of the columns of N matrices and allows a divorce from a matricized representation and a very consise expression of the PARAFAC decomposition. We explore the properties of the Tucker and Kruskal operators independently of the related decompositions. Additionally, we provide a review of the matrix and tensor operations that are frequently used in the context of tensor decompositions.

Modern applications such as Internet traffic, telecommunication records, and large-scale social networks generate massive amounts of data with multiple aspects and high dimensionalities. Tensors (i.e., multi-way arrays) provide a natural representation for such data. Consequently, tensor decompositions such as Tucker become important tools for summarization and analysis. One major challenge is how to deal with highdimensional, sparse data. In other words, how do we compute decompositions of tensors where most of the entries of the tensor are zero. Specialized techniques are needed for computing the Tucker decompositions for sparse tensors because standard algorithms do not account for the sparsity of the data. As a result, a surprising phenomenon is observed by practitioners: Despite the fact that there is enough memory to store both the input tensors and the factorized output tensors, memory overflows occur during the tensor factorization process. To address this intermediate blowup problem, we propose Memory-Efficient Tucker (MET). Based on the available memory, MET adaptively selects the right execution strategy during the decomposition. We provide quantitative and qualitative evaluation of MET on real tensors. It achieves over 1000X space reduction without sacrificing speed; it also allows us to work with much larger tensors that were too big to handle before. Finally, we demonstrate a data mining case-study using MET. 1

Abstract. The Semantic Web fosters novel applications targeting a more efficient and satisfying exploitation of the data available on the web, e.g. faceted browsing of linked open data. Large amounts and high diversity of knowledge in the Semantic Web pose the challenging question of appropriate relevance ranking for producing fine-grained and rich descriptions of the available data, e.g. to guide the user along most promising knowledge aspects. Existing methods for graphbased authority ranking lack support for fine-grained latent coherence between resources and predicates (i.e. support for link semantics in the linked data model). In this paper, we present TripleRank, a novel approach for faceted authority ranking in the context of RDF knowledge bases. TripleRank captures the additional latent semantics of Semantic Web data by means of statistical methods in order to produce richer descriptions of the available data. We model the Semantic Web by a 3-dimensional tensor that enables the seamless representation of arbitrary semantic links. For the analysis of that model, we apply the PARAFAC decomposition, which can be seen as a multi-modal counterpart to Web authority ranking with HITS. The result are groupings of resources and predicates that characterize their authority and navigational (hub) properties with respect to identified topics. We have applied TripleRank to multiple data sets from the linked open data community and gathered encouraging feedback in a user evaluation where TripleRank results have been exploited in a faceted browsing scenario. 1

Abstract. The focus of web search is moving away from returning relevant documents towards returning structured data as results to user queries. A vital part in the architecture of search engines are link-based ranking algorithms, which however are targeted towards hypertext documents. Existing ranking algorithms for structured data, on the other hand, require manual input of a domain expert and are thus not applicable in cases where data integrated from a large number of sources exhibits enormous variance in vocabularies used. In such environments, the authority of data sources is an important signal that the ranking algorithm has to take into account. This paper presents algorithms for prioritising data returned by queries over web datasets expressed in RDF. We introduce the notion of naming authority which provides a correspondence between identifiers and the sources which can speak authoritatively for these identifiers. Our algorithm uses the original PageRank method to assign authority values to data sources based on a naming authority graph, and then propagates the authority values to identifiers referenced in the sources. We conduct performance and quality evaluations of the method on a large web dataset. Our method is schema-independent, requires no manual input, and has applications in search, query processing, reasoning, and user interfaces over integrated datasets. 1

The blogosphere---the totality of blog-related Web sites--- has become a great source of trend analysis in areas such as product survey, customer relationship, and marketing. Existing approaches are based on simple counts, such as the number of entries or the number of links. In this paper, we introduce a novel concept, coined eigen-trend, to represent the temporal trend in a group of blogs with common interests and propose two new techniques for extracting eigentrends in blogs. First, we propose a trend analysis technique based on the singular value decomposition. Extracted eigentrends provide new insights into multiple trends on the same keyword. Second, we propose another trend analysis technique based on a higher-order singular value decomposition. This analyzes the blogosphere as a dynamic graph structure and extracts eigen-trends that reflect the structural changes of the blogosphere over time. Experimental studies based on synthetic data sets and a real blog data set show that our new techniques can reveal a lot of interesting trend information and insights in the blogosphere that are not obtainable from traditional count-based methods.

Data stream values are often associated with multiple aspects. For example, each value from environmental sensors may have an associated type (e.g., temperature, humidity, etc) as well as location. Aside from timestamp, type and location are the two additional aspects. How to model such streams? How to simultaneously find patterns within and across the multiple aspects? How to do it incrementally in a streaming fashion? In this paper, all these problems are addressed through a general data model, tensor streams, and an effective algorithmic framework, window-based tensor analysis (WTA). Two variations of WTA, independentwindow tensor analysis (IW) and moving-window tensor analysis (MW), are presented and evaluated extensively on real datasets. Finally, we illustrate one important application, Multi-Aspect Correlation Analysis (MACA), which uses WTA and we demonstrate its effectiveness on an environmental monitoring application. 1

The data in many disciplines such as social networks, web analysis, etc. is link-based, and the link structure can be exploited for many different data mining tasks. In this paper, we consider the problem of temporal link prediction: Given link data for time periods 1 through T, can we predict the links in time period T +1? Specifically, we look at bipartite graphs changing over time and consider matrix- and tensorbased methods for predicting links. We present a weight-based method for collapsing multi-year data into a single matrix. We show how the well-known Katz method for link prediction can be extended to bipartite graphs and, moreover, approximated in a scalable way using a truncated singular value decomposition. Using a CANDECOMP/PARAFAC tensor decomposition of the data, we illustrate the usefulness of exploiting the natural threedimensional structure of temporal link data. Through several numerical experiments, we demonstrate that both matrixand tensor-based techniques are effective for temporal link prediction despite the inherent difficulty of the problem.