Data modeling with the elliptical gamma distribution

Suvrit Sra, Reshad Hosseini, Lucas Theis, Matthias Bethge
Proceedings of the Eighteenth International Conference on Artificial Intelligence and Statistics, PMLR 38:903-911, 2015.

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.

Cite this Paper

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.} }
Endnote
%0 Conference Paper %T Data modeling with the elliptical gamma distribution %A Suvrit Sra %A Reshad Hosseini %A Lucas Theis %A Matthias Bethge %B Proceedings of the Eighteenth International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2015 %E Guy Lebanon %E S. V. N. Vishwanathan %F pmlr-v38-sra15 %I PMLR %P 903--911 %U https://proceedings.mlr.press/v38/sra15.html %V 38 %X 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.
RIS
TY - CPAPER TI - Data modeling with the elliptical gamma distribution AU - Suvrit Sra AU - Reshad Hosseini AU - Lucas Theis AU - Matthias Bethge BT - Proceedings of the Eighteenth International Conference on Artificial Intelligence and Statistics DA - 2015/02/21 ED - Guy Lebanon ED - S. V. N. Vishwanathan ID - pmlr-v38-sra15 PB - PMLR DP - Proceedings of Machine Learning Research VL - 38 SP - 903 EP - 911 L1 - http://proceedings.mlr.press/v38/sra15.pdf UR - https://proceedings.mlr.press/v38/sra15.html AB - 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. ER -
APA
Sra, S., Hosseini, R., Theis, L. & Bethge, M.. (2015). Data modeling with the elliptical gamma distribution. Proceedings of the Eighteenth International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 38:903-911 Available from https://proceedings.mlr.press/v38/sra15.html.

Related Material