Learning Globally Smooth Functions on Manifolds

Juan Cervino, Luiz F. O. Chamon, Benjamin David Haeffele, Rene Vidal, Alejandro Ribeiro
Proceedings of the 40th International Conference on Machine Learning, PMLR 202:3815-3854, 2023.

Abstract

Smoothness and low dimensional structures play central roles in improving generalization and stability in learning and statistics. This work combines techniques from semi-infinite constrained learning and manifold regularization to learn representations that are globally smooth on a manifold. To do so, it shows that under typical conditions the problem of learning a Lipschitz continuous function on a manifold is equivalent to a dynamically weighted manifold regularization problem. This observation leads to a practical algorithm based on a weighted Laplacian penalty whose weights are adapted using stochastic gradient techniques. It is shown that under mild conditions, this method estimates the Lipschitz constant of the solution, learning a globally smooth solution as a byproduct. Experiments on real world data illustrate the advantages of the proposed method relative to existing alternatives. Our code is available at https://github.com/JuanCervino/smoothbench.

Cite this Paper

BibTeX
@InProceedings{pmlr-v202-cervino23a, title = {Learning Globally Smooth Functions on Manifolds}, author = {Cervino, Juan and Chamon, Luiz F. O. and Haeffele, Benjamin David and Vidal, Rene and Ribeiro, Alejandro}, booktitle = {Proceedings of the 40th International Conference on Machine Learning}, pages = {3815--3854}, year = {2023}, editor = {Krause, Andreas and Brunskill, Emma and Cho, Kyunghyun and Engelhardt, Barbara and Sabato, Sivan and Scarlett, Jonathan}, volume = {202}, series = {Proceedings of Machine Learning Research}, month = {23--29 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v202/cervino23a/cervino23a.pdf}, url = {https://proceedings.mlr.press/v202/cervino23a.html}, abstract = {Smoothness and low dimensional structures play central roles in improving generalization and stability in learning and statistics. This work combines techniques from semi-infinite constrained learning and manifold regularization to learn representations that are globally smooth on a manifold. To do so, it shows that under typical conditions the problem of learning a Lipschitz continuous function on a manifold is equivalent to a dynamically weighted manifold regularization problem. This observation leads to a practical algorithm based on a weighted Laplacian penalty whose weights are adapted using stochastic gradient techniques. It is shown that under mild conditions, this method estimates the Lipschitz constant of the solution, learning a globally smooth solution as a byproduct. Experiments on real world data illustrate the advantages of the proposed method relative to existing alternatives. Our code is available at https://github.com/JuanCervino/smoothbench.} }
Endnote
%0 Conference Paper %T Learning Globally Smooth Functions on Manifolds %A Juan Cervino %A Luiz F. O. Chamon %A Benjamin David Haeffele %A Rene Vidal %A Alejandro Ribeiro %B Proceedings of the 40th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2023 %E Andreas Krause %E Emma Brunskill %E Kyunghyun Cho %E Barbara Engelhardt %E Sivan Sabato %E Jonathan Scarlett %F pmlr-v202-cervino23a %I PMLR %P 3815--3854 %U https://proceedings.mlr.press/v202/cervino23a.html %V 202 %X Smoothness and low dimensional structures play central roles in improving generalization and stability in learning and statistics. This work combines techniques from semi-infinite constrained learning and manifold regularization to learn representations that are globally smooth on a manifold. To do so, it shows that under typical conditions the problem of learning a Lipschitz continuous function on a manifold is equivalent to a dynamically weighted manifold regularization problem. This observation leads to a practical algorithm based on a weighted Laplacian penalty whose weights are adapted using stochastic gradient techniques. It is shown that under mild conditions, this method estimates the Lipschitz constant of the solution, learning a globally smooth solution as a byproduct. Experiments on real world data illustrate the advantages of the proposed method relative to existing alternatives. Our code is available at https://github.com/JuanCervino/smoothbench.
APA
Cervino, J., Chamon, L.F.O., Haeffele, B.D., Vidal, R. & Ribeiro, A.. (2023). Learning Globally Smooth Functions on Manifolds. Proceedings of the 40th International Conference on Machine Learning, in Proceedings of Machine Learning Research 202:3815-3854 Available from https://proceedings.mlr.press/v202/cervino23a.html.

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