  We study two practically important cases of model based clustering using
Gaussian Mixture Models: (1) when there is misspecification and (2) on high
dimensional data, in the light of recent advances in Gradient Descent (GD)
based optimization using Automatic Differentiation (AD). Our simulation studies
show that EM has better clustering performance, measured by Adjusted Rand
Index, compared to GD in cases of misspecification, whereas on high dimensional
data GD outperforms EM. We observe that both with EM and GD there are many
solutions with high likelihood but poor cluster interpretation. To address this
problem we design a new penalty term for the likelihood based on the Kullback
Leibler divergence between pairs of fitted components. Closed form expressions
for the gradients of this penalized likelihood are difficult to derive but AD
can be done effortlessly, illustrating the advantage of AD-based optimization.
Extensions of this penalty for high dimensional data and for model selection
are discussed. Numerical experiments on synthetic and real datasets demonstrate
the efficacy of clustering using the proposed penalized likelihood approach.
