Vector Quantile Regression on Manifolds

Marco Pegoraro, Sanketh Vedula, Aviv A Rosenberg, Irene Tallini, Emanuele Rodola, Alex Bronstein
Proceedings of The 27th International Conference on Artificial Intelligence and Statistics, PMLR 238:1999-2007, 2024.

Abstract

Quantile regression (QR) is a statistical tool for distribution-free estimation of conditional quantiles of a target variable given explanatory features. QR is limited by the assumption that the target distribution is univariate and defined on an Euclidean domain. Although the notion of quantiles was recently extended to multi-variate distributions, QR for multi-variate distributions on manifolds remains underexplored, even though many important applications inherently involve data distributed on, e.g., spheres (climate and geological phenomena), and tori (dihedral angles in proteins). By leveraging optimal transport theory and c-concave functions, we meaningfully define conditional vector quantile functions of high-dimensional variables on manifolds (M-CVQFs). Our approach allows for quantile estimation, regression, and computation of conditional confidence sets and likelihoods. We demonstrate the approach’s efficacy and provide insights regarding the meaning of non-Euclidean quantiles through synthetic and real data experiments.

Cite this Paper

BibTeX
@InProceedings{pmlr-v238-pegoraro24a, title = {Vector Quantile Regression on Manifolds}, author = {Pegoraro, Marco and Vedula, Sanketh and A Rosenberg, Aviv and Tallini, Irene and Rodola, Emanuele and Bronstein, Alex}, booktitle = {Proceedings of The 27th International Conference on Artificial Intelligence and Statistics}, pages = {1999--2007}, year = {2024}, editor = {Dasgupta, Sanjoy and Mandt, Stephan and Li, Yingzhen}, volume = {238}, series = {Proceedings of Machine Learning Research}, month = {02--04 May}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v238/pegoraro24a/pegoraro24a.pdf}, url = {https://proceedings.mlr.press/v238/pegoraro24a.html}, abstract = {Quantile regression (QR) is a statistical tool for distribution-free estimation of conditional quantiles of a target variable given explanatory features. QR is limited by the assumption that the target distribution is univariate and defined on an Euclidean domain. Although the notion of quantiles was recently extended to multi-variate distributions, QR for multi-variate distributions on manifolds remains underexplored, even though many important applications inherently involve data distributed on, e.g., spheres (climate and geological phenomena), and tori (dihedral angles in proteins). By leveraging optimal transport theory and c-concave functions, we meaningfully define conditional vector quantile functions of high-dimensional variables on manifolds (M-CVQFs). Our approach allows for quantile estimation, regression, and computation of conditional confidence sets and likelihoods. We demonstrate the approach’s efficacy and provide insights regarding the meaning of non-Euclidean quantiles through synthetic and real data experiments.} }
Endnote
%0 Conference Paper %T Vector Quantile Regression on Manifolds %A Marco Pegoraro %A Sanketh Vedula %A Aviv A Rosenberg %A Irene Tallini %A Emanuele Rodola %A Alex Bronstein %B Proceedings of The 27th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2024 %E Sanjoy Dasgupta %E Stephan Mandt %E Yingzhen Li %F pmlr-v238-pegoraro24a %I PMLR %P 1999--2007 %U https://proceedings.mlr.press/v238/pegoraro24a.html %V 238 %X Quantile regression (QR) is a statistical tool for distribution-free estimation of conditional quantiles of a target variable given explanatory features. QR is limited by the assumption that the target distribution is univariate and defined on an Euclidean domain. Although the notion of quantiles was recently extended to multi-variate distributions, QR for multi-variate distributions on manifolds remains underexplored, even though many important applications inherently involve data distributed on, e.g., spheres (climate and geological phenomena), and tori (dihedral angles in proteins). By leveraging optimal transport theory and c-concave functions, we meaningfully define conditional vector quantile functions of high-dimensional variables on manifolds (M-CVQFs). Our approach allows for quantile estimation, regression, and computation of conditional confidence sets and likelihoods. We demonstrate the approach’s efficacy and provide insights regarding the meaning of non-Euclidean quantiles through synthetic and real data experiments.
APA
Pegoraro, M., Vedula, S., A Rosenberg, A., Tallini, I., Rodola, E. & Bronstein, A.. (2024). Vector Quantile Regression on Manifolds. Proceedings of The 27th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 238:1999-2007 Available from https://proceedings.mlr.press/v238/pegoraro24a.html.

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