Generalized Multilinear Model for Dimensionality Reduction of Binary Tensors

Generalized multilinear model for dimensionality reduction of binary tensors (GMM-DR-BT) is a technique for computing low-rank approximations of multi-dimensional data objects, tensors. The model exposes a latent structure that represents dominant trends in the binary tensorial data while retaining as much information as possible. Recently, there exist several models for computing the low-rank approximations of tensors but to the best of our knowledge at present there is no principled framework for binary tensors. Although the binary tensors occur in many real world applications such as gait recognition, document analysis or graph mining. In the GMM-DR-BT model formulation, to account for binary nature of the data, each tensor element is modeled by a Bernoulli noise distribution. To extract the dominant trends in the data, the natural parameters of the Bernoulli distributions are constrained by employing the Tucker model to lie in a sub-space spanned by a reduced set of basis (principal) tensors. Bernoulli distribution is a member of exponential family with helpful analytical properties that allow us to derive an iterative scheme for estimation of the basis tensors and other model parameters via maximum likelihood. Furthermore, we extended the fully unsupervised GMMDR-BT model to the semi-supervised setting by forcing the model to search for a natural parameter subspace that represents a user specified compromise between modelling the quality and the degree of class separation. ∗Recommended by thesis supervisor: Dr. Peter Tiňo. Defended at Faculty of Informatics and Information Technologies, Slovak University of Technology in Bratislava on August 30, 2012. c © Copyright 2012. All rights reserved. Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies show this notice on the first page or initial screen of a display along with the full citation. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, to republish, to post on servers, to redistribute to lists, or to use any component of this work in other works requires prior specific permission and/or a fee. Permissions may be requested from STU Press, Vazovova 5, 811 07 Bratislava, Slovakia. Mažgut, J. Generalized Multilinear Model for Dimensionality Reduction of Binary Tensors. Information Sciences and Technologies Bulletin of the ACM Slovakia, Vol. 4, No. 3 (2012) 35-46 We evaluated and compared the proposed GMM-DR-BT technique with existing real-valued and nonnegative tensor decomposition methods in several experiments involving variety of high-dimensional datasets. The results suggest that the GMM-DR-BT model is better suited for modeling binary tensors than its real-valued and nonnegative counterparts.