Square Root Graphical Models: Multivariate Generalizations of Univariate Exponential Families that Permit Positive Dependencies

David Inouye, Pradeep Ravikumar, Inderjit Dhillon
Proceedings of The 33rd International Conference on Machine Learning, PMLR 48:2445-2453, 2016.

Abstract

We develop Square Root Graphical Models (SQR), a novel class of parametric graphical models that provides multivariate generalizations of univariate exponential family distributions. Previous multivariate graphical models [Yang et al. 2015] did not allow positive dependencies for the exponential and Poisson generalizations. However, in many real-world datasets, variables clearly have positive dependencies. For example, the airport delay time in New York—modeled as an exponential distribution—is positively related to the delay time in Boston. With this motivation, we give an example of our model class derived from the univariate exponential distribution that allows for almost arbitrary positive and negative dependencies with only a mild condition on the parameter matrix—a condition akin to the positive definiteness of the Gaussian covariance matrix. Our Poisson generalization allows for both positive and negative dependencies without any constraints on the parameter values. We also develop parameter estimation methods using node-wise regressions with \ell_1 regularization and likelihood approximation methods using sampling. Finally, we demonstrate our exponential generalization on a synthetic dataset and a real-world dataset of airport delay times.

Cite this Paper

BibTeX
@InProceedings{pmlr-v48-inouye16, title = {Square Root Graphical Models: Multivariate Generalizations of Univariate Exponential Families that Permit Positive Dependencies}, author = {Inouye, David and Ravikumar, Pradeep and Dhillon, Inderjit}, booktitle = {Proceedings of The 33rd International Conference on Machine Learning}, pages = {2445--2453}, year = {2016}, editor = {Balcan, Maria Florina and Weinberger, Kilian Q.}, volume = {48}, series = {Proceedings of Machine Learning Research}, address = {New York, New York, USA}, month = {20--22 Jun}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v48/inouye16.pdf}, url = {https://proceedings.mlr.press/v48/inouye16.html}, abstract = {We develop Square Root Graphical Models (SQR), a novel class of parametric graphical models that provides multivariate generalizations of univariate exponential family distributions. Previous multivariate graphical models [Yang et al. 2015] did not allow positive dependencies for the exponential and Poisson generalizations. However, in many real-world datasets, variables clearly have positive dependencies. For example, the airport delay time in New York—modeled as an exponential distribution—is positively related to the delay time in Boston. With this motivation, we give an example of our model class derived from the univariate exponential distribution that allows for almost arbitrary positive and negative dependencies with only a mild condition on the parameter matrix—a condition akin to the positive definiteness of the Gaussian covariance matrix. Our Poisson generalization allows for both positive and negative dependencies without any constraints on the parameter values. We also develop parameter estimation methods using node-wise regressions with \ell_1 regularization and likelihood approximation methods using sampling. Finally, we demonstrate our exponential generalization on a synthetic dataset and a real-world dataset of airport delay times.} }
Endnote
%0 Conference Paper %T Square Root Graphical Models: Multivariate Generalizations of Univariate Exponential Families that Permit Positive Dependencies %A David Inouye %A Pradeep Ravikumar %A Inderjit Dhillon %B Proceedings of The 33rd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2016 %E Maria Florina Balcan %E Kilian Q. Weinberger %F pmlr-v48-inouye16 %I PMLR %P 2445--2453 %U https://proceedings.mlr.press/v48/inouye16.html %V 48 %X We develop Square Root Graphical Models (SQR), a novel class of parametric graphical models that provides multivariate generalizations of univariate exponential family distributions. Previous multivariate graphical models [Yang et al. 2015] did not allow positive dependencies for the exponential and Poisson generalizations. However, in many real-world datasets, variables clearly have positive dependencies. For example, the airport delay time in New York—modeled as an exponential distribution—is positively related to the delay time in Boston. With this motivation, we give an example of our model class derived from the univariate exponential distribution that allows for almost arbitrary positive and negative dependencies with only a mild condition on the parameter matrix—a condition akin to the positive definiteness of the Gaussian covariance matrix. Our Poisson generalization allows for both positive and negative dependencies without any constraints on the parameter values. We also develop parameter estimation methods using node-wise regressions with \ell_1 regularization and likelihood approximation methods using sampling. Finally, we demonstrate our exponential generalization on a synthetic dataset and a real-world dataset of airport delay times.
RIS
TY - CPAPER TI - Square Root Graphical Models: Multivariate Generalizations of Univariate Exponential Families that Permit Positive Dependencies AU - David Inouye AU - Pradeep Ravikumar AU - Inderjit Dhillon BT - Proceedings of The 33rd International Conference on Machine Learning DA - 2016/06/11 ED - Maria Florina Balcan ED - Kilian Q. Weinberger ID - pmlr-v48-inouye16 PB - PMLR DP - Proceedings of Machine Learning Research VL - 48 SP - 2445 EP - 2453 L1 - http://proceedings.mlr.press/v48/inouye16.pdf UR - https://proceedings.mlr.press/v48/inouye16.html AB - We develop Square Root Graphical Models (SQR), a novel class of parametric graphical models that provides multivariate generalizations of univariate exponential family distributions. Previous multivariate graphical models [Yang et al. 2015] did not allow positive dependencies for the exponential and Poisson generalizations. However, in many real-world datasets, variables clearly have positive dependencies. For example, the airport delay time in New York—modeled as an exponential distribution—is positively related to the delay time in Boston. With this motivation, we give an example of our model class derived from the univariate exponential distribution that allows for almost arbitrary positive and negative dependencies with only a mild condition on the parameter matrix—a condition akin to the positive definiteness of the Gaussian covariance matrix. Our Poisson generalization allows for both positive and negative dependencies without any constraints on the parameter values. We also develop parameter estimation methods using node-wise regressions with \ell_1 regularization and likelihood approximation methods using sampling. Finally, we demonstrate our exponential generalization on a synthetic dataset and a real-world dataset of airport delay times. ER -
APA
Inouye, D., Ravikumar, P. & Dhillon, I.. (2016). Square Root Graphical Models: Multivariate Generalizations of Univariate Exponential Families that Permit Positive Dependencies. Proceedings of The 33rd International Conference on Machine Learning, in Proceedings of Machine Learning Research 48:2445-2453 Available from https://proceedings.mlr.press/v48/inouye16.html.

Related Material