Data modeling with the elliptical gamma distribution
Suvrit Sra, Reshad Hosseini, Lucas Theis, Matthias Bethge
Abstract
We study mixture modeling using the elliptical gamma (EG) distribution, a non-Gaussian distribution that allows heavy and light tail and peak behaviors. We first consider maximum likelihood parameter estimation, a task that turns out to be very challenging: we must handle positive definiteness constraints, and more crucially, we must handle possibly nonconcave log-likelihoods, which makes maximization hard. We overcome these difficulties by developing algorithms based on fixed-point theory; our methods respect the psd constraint, while also efficiently solving the (possibly) nonconcave maximization to global optimality. Subsequently, we focus on mixture modeling using EG distributions: we present a closed-form expression of the KL-divergence between two EG distributions, which we then combine with our ML estimation methods to obtain an efficient split-and-merge expectation maximization algorithm. We illustrate the use of our model and algorithms on a dataset of natural image patches.
BibTeX
@InProceedings{pmlr-v38-sra15,
title = {{Data modeling with the elliptical gamma distribution}},
author = {Sra, Suvrit and Hosseini, Reshad and Theis, Lucas and Bethge, Matthias},
booktitle = {Proceedings of the Eighteenth International Conference on Artificial Intelligence and Statistics},
pages = {903--911},
year = {2015},
editor = {Lebanon, Guy and Vishwanathan, S. V. N.},
volume = {38},
series = {Proceedings of Machine Learning Research},
address = {San Diego, California, USA},
month = {09--12 May},
publisher = {PMLR},
pdf = {http://proceedings.mlr.press/v38/sra15.pdf},
url = {https://proceedings.mlr.press/v38/sra15.html},
abstract = {We study mixture modeling using the elliptical gamma (EG) distribution, a non-Gaussian distribution that allows heavy and light tail and peak behaviors. We first consider maximum likelihood parameter estimation, a task that turns out to be very challenging: we must handle positive definiteness constraints, and more crucially, we must handle possibly nonconcave log-likelihoods, which makes maximization hard. We overcome these difficulties by developing algorithms based on fixed-point theory; our methods respect the psd constraint, while also efficiently solving the (possibly) nonconcave maximization to global optimality. Subsequently, we focus on mixture modeling using EG distributions: we present a closed-form expression of the KL-divergence between two EG distributions, which we then combine with our ML estimation methods to obtain an efficient split-and-merge expectation maximization algorithm. We illustrate the use of our model and algorithms on a dataset of natural image patches.}
}