Approximating the covariance matrix of GMMs with low-rank perturbations

Abstract

Covariance matrices capture correlations that are invaluable in modeling 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 modeling 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.