Linearized Wasserstein Barycenters: Synthesis, Analysis, Representational Capacity, and Applications

Matthew Werenski, Brendan Mallery, Shuchin Aeron, James M. Murphy
Proceedings of The 28th International Conference on Artificial Intelligence and Statistics, PMLR 258:4555-4563, 2025.

Abstract

We propose the linear barycentric coding model (LBCM) which utilizes the linear optimal transport (LOT) metric for analysis and synthesis of probability measures. We provide a closed-form solution to the variational problem characterizing the probability measures in the LBCM and establish equivalence of the LBCM to the set of 2-Wasserstein barycenters in the special case of compatible measures. Computational methods for synthesizing and analyzing measures in the LBCM are developed with finite sample guarantees. One of our main theoretical contributions is to identify an LBCM, expressed in terms of a simple family, which is sufficient to express all probability measures on the closed unit interval. We show that a natural analogous construction of an LBCM in 2 dimensions fails, and we leave it as an open problem to identify the proper extension in more than 1 dimension. We conclude by demonstrating the utility of LBCM for covariance estimation and data imputation.

Cite this Paper

BibTeX
@InProceedings{pmlr-v258-werenski25a, title = {Linearized Wasserstein Barycenters: Synthesis, Analysis, Representational Capacity, and Applications}, author = {Werenski, Matthew and Mallery, Brendan and Aeron, Shuchin and Murphy, James M.}, booktitle = {Proceedings of The 28th International Conference on Artificial Intelligence and Statistics}, pages = {4555--4563}, year = {2025}, editor = {Li, Yingzhen and Mandt, Stephan and Agrawal, Shipra and Khan, Emtiyaz}, volume = {258}, series = {Proceedings of Machine Learning Research}, month = {03--05 May}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v258/main/assets/werenski25a/werenski25a.pdf}, url = {https://proceedings.mlr.press/v258/werenski25a.html}, abstract = {We propose the linear barycentric coding model (LBCM) which utilizes the linear optimal transport (LOT) metric for analysis and synthesis of probability measures. We provide a closed-form solution to the variational problem characterizing the probability measures in the LBCM and establish equivalence of the LBCM to the set of 2-Wasserstein barycenters in the special case of compatible measures. Computational methods for synthesizing and analyzing measures in the LBCM are developed with finite sample guarantees. One of our main theoretical contributions is to identify an LBCM, expressed in terms of a simple family, which is sufficient to express all probability measures on the closed unit interval. We show that a natural analogous construction of an LBCM in 2 dimensions fails, and we leave it as an open problem to identify the proper extension in more than 1 dimension. We conclude by demonstrating the utility of LBCM for covariance estimation and data imputation.} }
Endnote
%0 Conference Paper %T Linearized Wasserstein Barycenters: Synthesis, Analysis, Representational Capacity, and Applications %A Matthew Werenski %A Brendan Mallery %A Shuchin Aeron %A James M. Murphy %B Proceedings of The 28th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2025 %E Yingzhen Li %E Stephan Mandt %E Shipra Agrawal %E Emtiyaz Khan %F pmlr-v258-werenski25a %I PMLR %P 4555--4563 %U https://proceedings.mlr.press/v258/werenski25a.html %V 258 %X We propose the linear barycentric coding model (LBCM) which utilizes the linear optimal transport (LOT) metric for analysis and synthesis of probability measures. We provide a closed-form solution to the variational problem characterizing the probability measures in the LBCM and establish equivalence of the LBCM to the set of 2-Wasserstein barycenters in the special case of compatible measures. Computational methods for synthesizing and analyzing measures in the LBCM are developed with finite sample guarantees. One of our main theoretical contributions is to identify an LBCM, expressed in terms of a simple family, which is sufficient to express all probability measures on the closed unit interval. We show that a natural analogous construction of an LBCM in 2 dimensions fails, and we leave it as an open problem to identify the proper extension in more than 1 dimension. We conclude by demonstrating the utility of LBCM for covariance estimation and data imputation.
APA
Werenski, M., Mallery, B., Aeron, S. & Murphy, J.M.. (2025). Linearized Wasserstein Barycenters: Synthesis, Analysis, Representational Capacity, and Applications. Proceedings of The 28th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 258:4555-4563 Available from https://proceedings.mlr.press/v258/werenski25a.html.

Related Material