The most common approaches for estimating multivariate density assume a parametric form for the joint distribution. The choice of this parametric form imposes constraints on the marginal distributions. Copula models disentangle the choice of marginals from the joint distributions, making it a powerful model for multivariate density estimation. However, so far, they have been widely studied mostly for low dimensional multivariate. In this paper, we investigate a popular Copula model - the Gaussian Copula model - for high dimensional settings. They however require estimation of a full correlation matrix which can cause data scarcity in this setting. One approach to address this problem is to impose constraints on the parameter space. In this paper, we present Toeplitz correlation structure to reduce the number of Gaussian Copula parameter. To increase the flexibility of our model, we also introduce mixture of Gaussian Copula as a natural extension of the Gaussian Copula model. Through empirical evaluation of likelihood on held-out data, we study the trade-off between correlation constraints and mixture flexibility, and report results on wine data sets from the UCI Repository as well as our corpus of monkey vocalizations. We find that mixture of Gaussian Copula with Toeplitz correlation structure models the data consistently better than Gaussian mixture models with equivalent number of parameters.
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