Title: Approximating the covariance matrix of GMMs with low-rank perturbations

Authors: Malik Magdon-Ismail; Jonathan T. Purnell

Addresses: Department of Computer Science, Rensselaer Polytechnic Institute, Troy, NY 12180, USA. ' Department of Computer Science, Rensselaer Polytechnic Institute, Troy, NY 12180, USA

Abstract: Covariance matrices capture correlations that are invaluable in modelling real-life datasets. Using all d² elements of the covariance (in d dimensions) is costly and could result in over-fitting; and the simple diagonal approximation can be over-restrictive. In this work, we present a new model, the low-rank Gaussian mixture model (LRGMM), for modelling data which can be extended to identifying partitions or overlapping clusters. The curse of dimensionality that arises in calculating the covariance matrices of the GMM is countered by using low-rank perturbed diagonal matrices. The efficiency is comparable to the diagonal approximation, yet one can capture correlations among the dimensions. Our experiments reveal the LRGMM to be an efficient and highly applicable tool for working with large high-dimensional datasets.

Keywords: Gaussian mixture models; Carl Friedrich Gauss; maximum likelihood; expectation-maximisation; covariance matrix; low-rank perturbations; overlapping clusters; large datasets; real-life datasets; over-fitting; diagonal approximations; low-rank models; partitions; dimensionality; low-rank matrices; perturbed diagonal matrices; efficiency; correlations; high-dimensional datasets; data mining; data modelling; data management; intelligent data analysis.

DOI: 10.1504/IJDMMM.2012.046805

International Journal of Data Mining, Modelling and Management, 2012 Vol.4 No.2, pp.107 - 122

Published online: 23 Aug 2014 *

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