Tails of Lipschitz Triangular Flows

Priyank Jaini, Ivan Kobyzev, Yaoliang Yu, Marcus Brubaker
Proceedings of the 37th International Conference on Machine Learning, PMLR 119:4673-4681, 2020.

Abstract

We investigate the ability of popular flow models to capture tail-properties of a target density by studying the increasing triangular maps used in these flow methods acting on a tractable source density. We show that the density quantile functions of the source and target density provide a precise characterization of the slope of transformation required to capture tails in a target density. We further show that any Lipschitz-continuous transport map acting on a source density will result in a density with similar tail properties as the source, highlighting the trade-off between the importance of choosing a complex source density and a sufficiently expressive transformation to capture desirable properties of a target density. Subsequently, we illustrate that flow models like Real-NVP, MAF, and Glow as implemented lack the ability to capture a distribution with non-Gaussian tails. We circumvent this problem by proposing tail-adaptive flows consisting of a source distribution that can be learned simultaneously with the triangular map to capture tail-properties of a target density. We perform several synthetic and real-world experiments to complement our theoretical findings.

Cite this Paper

BibTeX
@InProceedings{pmlr-v119-jaini20a, title = {Tails of {L}ipschitz Triangular Flows}, author = {Jaini, Priyank and Kobyzev, Ivan and Yu, Yaoliang and Brubaker, Marcus}, booktitle = {Proceedings of the 37th International Conference on Machine Learning}, pages = {4673--4681}, year = {2020}, editor = {III, Hal Daumé and Singh, Aarti}, volume = {119}, series = {Proceedings of Machine Learning Research}, month = {13--18 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v119/jaini20a/jaini20a.pdf}, url = {https://proceedings.mlr.press/v119/jaini20a.html}, abstract = {We investigate the ability of popular flow models to capture tail-properties of a target density by studying the increasing triangular maps used in these flow methods acting on a tractable source density. We show that the density quantile functions of the source and target density provide a precise characterization of the slope of transformation required to capture tails in a target density. We further show that any Lipschitz-continuous transport map acting on a source density will result in a density with similar tail properties as the source, highlighting the trade-off between the importance of choosing a complex source density and a sufficiently expressive transformation to capture desirable properties of a target density. Subsequently, we illustrate that flow models like Real-NVP, MAF, and Glow as implemented lack the ability to capture a distribution with non-Gaussian tails. We circumvent this problem by proposing tail-adaptive flows consisting of a source distribution that can be learned simultaneously with the triangular map to capture tail-properties of a target density. We perform several synthetic and real-world experiments to complement our theoretical findings.} }
Endnote
%0 Conference Paper %T Tails of Lipschitz Triangular Flows %A Priyank Jaini %A Ivan Kobyzev %A Yaoliang Yu %A Marcus Brubaker %B Proceedings of the 37th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2020 %E Hal Daumé III %E Aarti Singh %F pmlr-v119-jaini20a %I PMLR %P 4673--4681 %U https://proceedings.mlr.press/v119/jaini20a.html %V 119 %X We investigate the ability of popular flow models to capture tail-properties of a target density by studying the increasing triangular maps used in these flow methods acting on a tractable source density. We show that the density quantile functions of the source and target density provide a precise characterization of the slope of transformation required to capture tails in a target density. We further show that any Lipschitz-continuous transport map acting on a source density will result in a density with similar tail properties as the source, highlighting the trade-off between the importance of choosing a complex source density and a sufficiently expressive transformation to capture desirable properties of a target density. Subsequently, we illustrate that flow models like Real-NVP, MAF, and Glow as implemented lack the ability to capture a distribution with non-Gaussian tails. We circumvent this problem by proposing tail-adaptive flows consisting of a source distribution that can be learned simultaneously with the triangular map to capture tail-properties of a target density. We perform several synthetic and real-world experiments to complement our theoretical findings.
APA
Jaini, P., Kobyzev, I., Yu, Y. & Brubaker, M.. (2020). Tails of Lipschitz Triangular Flows. Proceedings of the 37th International Conference on Machine Learning, in Proceedings of Machine Learning Research 119:4673-4681 Available from https://proceedings.mlr.press/v119/jaini20a.html.

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