Towards Optimal Transport with Global Invariances

David Alvarez-Melis, Stefanie Jegelka, Tommi S. Jaakkola
Proceedings of the Twenty-Second International Conference on Artificial Intelligence and Statistics, PMLR 89:1870-1879, 2019.

Abstract

Many problems in machine learning involve calculating correspondences between sets of objects, such as point clouds or images. Discrete optimal transport provides a natural and successful approach to such tasks whenever the two sets of objects can be represented in the same space, or at least distances between them can be directly evaluated. Unfortunately neither requirement is likely to hold when object representations are learned from data. Indeed, automatically derived representations such as word embeddings are typically fixed only up to some global transformations, for example, reflection or rotation. As a result, pairwise distances across two such instances are ill-defined without specifying their relative transformation. In this work, we propose a general framework for optimal transport in the presence of latent global transformations. We cast the problem as a joint optimization over transport couplings and transformations chosen from a flexible class of invariances, propose algorithms to solve it, and show promising results in various tasks, including a popular unsupervised word translation benchmark.

Cite this Paper

BibTeX
@InProceedings{pmlr-v89-alvarez-melis19a, title = {Towards Optimal Transport with Global Invariances}, author = {Alvarez-Melis, David and Jegelka, Stefanie and Jaakkola, Tommi S.}, booktitle = {Proceedings of the Twenty-Second International Conference on Artificial Intelligence and Statistics}, pages = {1870--1879}, year = {2019}, editor = {Chaudhuri, Kamalika and Sugiyama, Masashi}, volume = {89}, series = {Proceedings of Machine Learning Research}, month = {16--18 Apr}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v89/alvarez-melis19a/alvarez-melis19a.pdf}, url = {https://proceedings.mlr.press/v89/alvarez-melis19a.html}, abstract = {Many problems in machine learning involve calculating correspondences between sets of objects, such as point clouds or images. Discrete optimal transport provides a natural and successful approach to such tasks whenever the two sets of objects can be represented in the same space, or at least distances between them can be directly evaluated. Unfortunately neither requirement is likely to hold when object representations are learned from data. Indeed, automatically derived representations such as word embeddings are typically fixed only up to some global transformations, for example, reflection or rotation. As a result, pairwise distances across two such instances are ill-defined without specifying their relative transformation. In this work, we propose a general framework for optimal transport in the presence of latent global transformations. We cast the problem as a joint optimization over transport couplings and transformations chosen from a flexible class of invariances, propose algorithms to solve it, and show promising results in various tasks, including a popular unsupervised word translation benchmark.} }
Endnote
%0 Conference Paper %T Towards Optimal Transport with Global Invariances %A David Alvarez-Melis %A Stefanie Jegelka %A Tommi S. Jaakkola %B Proceedings of the Twenty-Second International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2019 %E Kamalika Chaudhuri %E Masashi Sugiyama %F pmlr-v89-alvarez-melis19a %I PMLR %P 1870--1879 %U https://proceedings.mlr.press/v89/alvarez-melis19a.html %V 89 %X Many problems in machine learning involve calculating correspondences between sets of objects, such as point clouds or images. Discrete optimal transport provides a natural and successful approach to such tasks whenever the two sets of objects can be represented in the same space, or at least distances between them can be directly evaluated. Unfortunately neither requirement is likely to hold when object representations are learned from data. Indeed, automatically derived representations such as word embeddings are typically fixed only up to some global transformations, for example, reflection or rotation. As a result, pairwise distances across two such instances are ill-defined without specifying their relative transformation. In this work, we propose a general framework for optimal transport in the presence of latent global transformations. We cast the problem as a joint optimization over transport couplings and transformations chosen from a flexible class of invariances, propose algorithms to solve it, and show promising results in various tasks, including a popular unsupervised word translation benchmark.
APA
Alvarez-Melis, D., Jegelka, S. & Jaakkola, T.S.. (2019). Towards Optimal Transport with Global Invariances. Proceedings of the Twenty-Second International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 89:1870-1879 Available from https://proceedings.mlr.press/v89/alvarez-melis19a.html.

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