Efficient MATLAB computations with sparse and factored tensors.

In this paper, the term tensor refers simply to a multidimensional or N-way array, and we consider how specially structured tensors allow for efficient storage and computation. First, we study sparse tensors, which have the property that the vast majority of the elements are zero. We propose storing sparse tensors using coordinate format and describe the computational efficiency of this scheme for various mathematical operations, including those typical to tensor decomposition algorithms. Second, we study factored tensors, which have the property that they can be assembled from more basic components. We consider two specific types: a Tucker tensor can be expressed as the product of a core tensor (which itself may be dense, sparse, or factored) and a matrix along each mode, and a Kruskal tensor can be expressed as the sum of rank-1 tensors. We are interested in the case where the storage of the components is less than the storage of the full tensor, and we demonstrate that many elementary operations can be computed using only the components. All of the efficiencies described in this paper are implemented in the Tensor Toolbox for MATLAB

Factorization of multiple tensors for supervised feature extraction

© Springer International Publishing AG 2016. Tensors are effective representations for complex and timevarying networks. The factorization of a tensor provides a high-quality low-rank compact basis for each dimension of the tensor, which facilitates the interpretation of important structures of the represented data. Many existing tensor factorization (TF) methods assume there is one tensor that needs to be decomposed to low-rank factors. However in practice, data are usually generated from different time periods or by different class labels, which are represented by a sequence of multiple tensors associated with different labels. When one needs to analyse and compare multiple tensors, existing TF methods are unsuitable for discovering all potentially useful patterns, as they usually fail to discover either common or unique factors among the tensors: (1) if each tensor is factorized separately, the factor matrices will fail to explicitly capture the common information shared by different tensors, and (2) if tensors are concatenated together to form a larger “overall” tensor and then factorize this concatenated tensor, the intrinsic unique subspaces that are specific to each tensor will be lost. The cause of such an issue is mainly from the fact that existing tensor factorization methods handle data observations in an unsupervised way, considering only features but not labels of the data. To tackle this problem, we design a novel probabilistic tensor factorization model that takes both features and class labels of tensors into account, and produces informative common and unique factors of all tensors simultaneously. Experiment results on feature extraction in classification problems demonstrate the effectiveness of the factors discovered by our method

Predicting Multi-actor collaborations using Hypergraphs

Social networks are now ubiquitous and most of them contain interactions involving multiple actors (groups) like author collaborations, teams or emails in an organizations, etc. Hypergraphs are natural structures to effectively capture multi-actor interactions which conventional dyadic graphs fail to capture. In this work the problem of predicting collaborations is addressed while modeling the collaboration network as a hypergraph network. The problem of predicting future multi-actor collaboration is mapped to hyperedge prediction problem. Given that the higher order edge prediction is an inherently hard problem, in this work we restrict to the task of predicting edges (collaborations) that have already been observed in past. In this work, we propose a novel use of hyperincidence temporal tensors to capture time varying hypergraphs and provides a tensor decomposition based prediction algorithm. We quantitatively compare the performance of the hypergraphs based approach with the conventional dyadic graph based approach. Our hypothesis that hypergraphs preserve the information that simple graphs destroy is corroborated by experiments using author collaboration network from the DBLP dataset. Our results demonstrate the strength of hypergraph based approach to predict higher order collaborations (size>4) which is very difficult using dyadic graph based approach. Moreover, while predicting collaborations of size>2 hypergraphs in most cases provide better results with an average increase of approx. 45% in F-Score for different sizes = {3,4,5,6,7}