Results 1  10
of
79
Tensor Decompositions and Applications
 SIAM REVIEW
, 2009
"... This survey provides an overview of higherorder tensor decompositions, their applications, and available software. A tensor is a multidimensional or N way array. Decompositions of higherorder tensors (i.e., N way arrays with N â¥ 3) have applications in psychometrics, chemometrics, signal proce ..."
Abstract

Cited by 705 (17 self)
 Add to MetaCart
(Show Context)
This survey provides an overview of higherorder tensor decompositions, their applications, and available software. A tensor is a multidimensional or N way array. Decompositions of higherorder tensors (i.e., N way arrays with N â¥ 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 higherorder extensions of the matrix singular value decompo
sition: CANDECOMP/PARAFAC (CP) decomposes a tensor as a sum of rankone tensors, and the Tucker decomposition is a higherorder 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 Nway Toolbox and Tensor Toolbox, both for MATLAB, and the Multilinear Engine are examples of software packages for working with tensors.
From frequency to meaning : Vector space models of semantics
 Journal of Artificial Intelligence Research
, 2010
"... Computers understand very little of the meaning of human language. This profoundly limits our ability to give instructions to computers, the ability of computers to explain their actions to us, and the ability of computers to analyse and process text. Vector space models (VSMs) of semantics are begi ..."
Abstract

Cited by 322 (3 self)
 Add to MetaCart
(Show Context)
Computers understand very little of the meaning of human language. This profoundly limits our ability to give instructions to computers, the ability of computers to explain their actions to us, and the ability of computers to analyse and process text. Vector space models (VSMs) of semantics are beginning to address these limits. This paper surveys the use of VSMs for semantic processing of text. We organize the literature on VSMs according to the structure of the matrix in a VSM. There are currently three broad classes of VSMs, based on term–document, word–context, and pair–pattern matrices, yielding three classes of applications. We survey a broad range of applications in these three categories and we take a detailed look at a specific open source project in each category. Our goal in this survey is to show the breadth of applications of VSMs for semantics, to provide a new perspective on VSMs for those who are already familiar with the area, and to provide pointers into the literature for those who are less familiar with the field. 1.
Statistical Performance of Convex Tensor Decomposition
"... We analyze the statistical performance of a recently proposed convex tensor decomposition algorithm. Conventionally tensor decomposition has been formulated as nonconvex optimization problems, which hindered the analysis of their performance. We show under some conditions that the mean squared erro ..."
Abstract

Cited by 35 (5 self)
 Add to MetaCart
(Show Context)
We analyze the statistical performance of a recently proposed convex tensor decomposition algorithm. Conventionally tensor decomposition has been formulated as nonconvex optimization problems, which hindered the analysis of their performance. We show under some conditions that the mean squared error of the convex method scales linearly with the quantity we call the normalized rank of the true tensor. The current analysis naturally extends the analysis of convex lowrank matrix estimation to tensors. Furthermore, we show through numerical experiments that our theory can precisely predict the scaling behaviour in practice. 1
Probabilistic models for incomplete multidimensional arrays
 In Proceedings of the 12th International Conference on Artificial Intelligence and Statistics
, 2009
"... In multiway data, each sample is measured by multiple sets of correlated attributes. We develop a probabilistic framework for modeling structural dependency from partially observed multidimensional array data, known as pTucker. Latent components associated with individual array dimensions are joint ..."
Abstract

Cited by 31 (2 self)
 Add to MetaCart
(Show Context)
In multiway data, each sample is measured by multiple sets of correlated attributes. We develop a probabilistic framework for modeling structural dependency from partially observed multidimensional array data, known as pTucker. Latent components associated with individual array dimensions are jointly retrieved while the core tensor is integrated out. The resulting algorithm is capable of handling largescale data sets. We verify the usefulness of this approach by comparing against classical models on applications to modeling amino acid fluorescence, collaborative filtering and a number of benchmark multiway array data. 1
Multilayer networks
 TOOL FOR MULTILAYER ANALYSIS AND VISUALIZATION OF NETWORKS 17 OF 18
, 2014
"... In most natural and engineered systems, a set of entities interact with each other in complicated patterns that can encompass multiple types of relationships, change in time, and include other types of complications. Such systems include multiple subsystems and layers of connectivity, and it is impo ..."
Abstract

Cited by 30 (7 self)
 Add to MetaCart
(Show Context)
In most natural and engineered systems, a set of entities interact with each other in complicated patterns that can encompass multiple types of relationships, change in time, and include other types of complications. Such systems include multiple subsystems and layers of connectivity, and it is important to take such “multilayer” features into account to try to improve our understanding of complex systems. Consequently, it is necessary to generalize “traditional ” network theory by developing (and validating) a framework and associated tools to study multilayer systems in a comprehensive fashion. The origins of such efforts date back several decades and arose in multiple disciplines, and now the study of multilayer networks has become one of the most important directions in network science. In this paper, we discuss the history of multilayer networks (and related concepts) and review the exploding body of work on such networks. To unify the disparate terminology in the large body of recent work, we discuss a general framework for multilayer networks, construct a dictionary
Allatonce Optimization for Coupled Matrix and Tensor Factorizations
, 1105
"... Joint analysis of data from multiple sources has the potential to improve our understanding of the underlying structures in complex data sets. For instance, in restaurant recommendation systems, recommendations can be based on rating histories of customers. In addition to rating histories, customers ..."
Abstract

Cited by 29 (3 self)
 Add to MetaCart
(Show Context)
Joint analysis of data from multiple sources has the potential to improve our understanding of the underlying structures in complex data sets. For instance, in restaurant recommendation systems, recommendations can be based on rating histories of customers. In addition to rating histories, customers ’ social networks (e.g., Facebook friendships) and restaurant categories information (e.g., Thai or Italian) can also be used to make better recommendations. The task of fusing data, however, is challenging since data sets can be incomplete and heterogeneous, i.e., data consist of both matrices, e.g., the person by person social network matrix or the restaurant by category matrix, and higherorder tensors, e.g., the “ratings ” tensor of the form restaurant by meal by person. In this paper, we are particularly interested in fusing data sets with the goal of capturing their underlying latent structures. We formulate this problem as a coupled matrix and tensor factorization (CMTF) problem where heterogeneous data sets are modeled by fitting outerproduct models to higherorder tensors and matrices in a coupled manner. Unlike traditional approaches solving this problem using alternating algorithms, we propose an allatonce optimization approach called CMTFOPT (CMTFOPTimization), which is a gradientbased optimization approach for joint analysis of matrices and higherorder tensors. We also extend the algorithm to handle coupled incomplete data sets. Using numerical experiments, we demonstrate that the proposed allatonce approach is more accurate than the alternating least squares approach.
A Unified Framework for Providing Recommendations in Social Tagging Systems Based on Ternary Semantic Analysis
"... 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, bas ..."
Abstract

Cited by 25 (4 self)
 Add to MetaCart
(Show Context)
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 3order tensor, on which multiway latent semantic analysis and dimensionality reduction is performed using both the Higher Order Singular Value Decomposition (HOSVD) method and the KernelSVD smoothing technique. We perform experimental comparison of the proposed method against stateoftheart 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
Scalable Tensor Factorizations with Missing Data
 SIAM INTERNATIONAL CONFERENCE ON DATA MINING
, 2010
"... The problem of missing data is ubiquitous in domains such as biomedical signal processing, network traffic analysis, bibliometrics, social network analysis, chemometrics, computer vision, and communication networksall domains in which data collection is subject to occasional errors. Moreover, the ..."
Abstract

Cited by 24 (1 self)
 Add to MetaCart
(Show Context)
The problem of missing data is ubiquitous in domains such as biomedical signal processing, network traffic analysis, bibliometrics, social network analysis, chemometrics, computer vision, and communication networksall domains in which data collection is subject to occasional errors. Moreover, these data sets can be quite large and have more than two axes of variation, e.g., sender, receiver, time. Many applications in those domains aim to capture the underlying latent structure of the data; in other words, they need to factorize data sets with missing entries. If we cannot address the problem of missing data, many important data sets will be discarded or improperly analyzed. Therefore, we need a robust and scalable approach for factorizing multiway arrays (i.e., tensors) in the presence of missing data. We focus on one of the most wellknown tensor factorizations, CANDECOMP/PARAFAC (CP), and formulate the CP model as a weighted least squares problem that models only the known entries. We develop an algorithm called CPWOPT (CP Weighted OPTimization) using a firstorder optimization approach to solve the weighted least squares problem. Based on extensive numerical experiments, our algorithm is shown to successfully factor tensors with noise and up to 70% missing data. Moreover, our approach is significantly faster than the leading alternative and scales to larger problems. To show the realworld usefulness of CPWOPT, we illustrate its applicability on a novel EEG (electroencephalogram) application where missing data is frequently encountered due to disconnections of electrodes.
Swamp reducing technique for tensor decompositions, submitted for publication
, 2008
"... There are numerous applications of tensor analysis in signal processing, such as, blind multiuser separationequalizationdetection and blind identification. As the applicability of tensor analysis widens, the numerical techniques must improve to accommodate new data. We present a new numerical me ..."
Abstract

Cited by 17 (5 self)
 Add to MetaCart
(Show Context)
There are numerous applications of tensor analysis in signal processing, such as, blind multiuser separationequalizationdetection and blind identification. As the applicability of tensor analysis widens, the numerical techniques must improve to accommodate new data. We present a new numerical method for tensor analysis. The method is based on the iterated Tikhonov regularization and a parameter choice rule. Together these elements dramatically accelerate the wellknown Alternating LeastSquares method. 1.
Link Prediction on Evolving Data using Matrix and Tensor Factorizations
 IEEE INTERNATIONAL CONFERENCE ON DATA MINING WORKSHOPS
, 2009
"... The data in many disciplines such as social networks, web analysis, etc. is linkbased, 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 l ..."
Abstract

Cited by 17 (1 self)
 Add to MetaCart
(Show Context)
The data in many disciplines such as social networks, web analysis, etc. is linkbased, 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 weightbased method for collapsing multiyear data into a single matrix. We show how the wellknown 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 tensorbased techniques are effective for temporal link prediction despite the inherent difficulty of the problem.