Extracting clusters from large datasets with multiple similarity measures.

We consider the problem of how to group information when multiple similarities are known. For a group of people, we may know their education, geographic location and family connections and want to cluster the people by treating all three of these similarities simultaneously. Our approach is to store each similarity as a slice in a tensor. The similarity measures are generated by comparing features. Generally, the object similarity matrix is dense. However it can be stored implicitly as the product of a sparse matrix, representing the object-feature matrix, and its transpose. For this new type of tensor where dense slices are stored implicitly, we have created a new decomposition called Implicit Slice Canonical Decomposition (IMSCAND). Our decomposition is equivalent to the tensor CANDECOMP/PARAFAC decomposition, which is a higher-order analogue of the matrix Singular Value decomposition (SVD) and Principal Component Analysis (PCA). From IMSCAND we obtain compilation feature vectors which are clustered using k-means. We demonstrate the applicability of IMSCAND on a set of journal articles with multiple similarities. 1. Introduction. Datasets naturally have mulitple similarities. For a group of people, we might know their age, education, geographic location, and social and family connections. For a set of published papers, we know the authors, citations, and terms in the abstract, title, and keywords. Even for a computer hard drive, we know the names of the files, their saved location, their time stamp, and their contents. For each of these examples, if we wanted to find a way to cluster the people, documents, or computer files, our approach is to treat all the similarities concurrently. For each similarity, a similarity matrix is formed, with objects (people, documents, files, etc.) as both the rows and columns. Each of the similarity matrices is a slice in a tensor, and a tensor decomposition is used to assemble the multiple adjacency matrices into a set of compilation feature vectors. These feature vectors can then be clustered using the k-means clustering algorithm. This approach was used by Dunlavy et al. (6). Our approach to this problem is special because of how we form the similarity matrices. In general, similarity matrices are dense, limiting the number of objects and features because of the large amount of work required to compute a decomposition for a dense tensor. We form the dense similarity matrices implicitly by storing only sparse object-feature matrices. For example, an article-author matrix is a sparse object-feature matrix with documents as rows and authors as columns of the matrix. Since most articles have just a few authors, we can see how this is a sparse matrix. Then the adjacency matrix is stored implicitly as the product of the sparse object-feature matrix and its transpose (a feature-object matrix). Our new decomposition is called Implicit Slice Canonical Decomposition (IMSCAND) and does all of its computations on the sparse matrices only. This allows us to treat much larger problems than if we stored the full similarity matrices. We illustrate the e ectiveness of our decomposition on a set of journal publications from the Society of Industrial and Applied Mathematics (SIAM). Our long-term goal is to use this approach to cluster files on computer hard drives.