Non-linear Embeddings in Hilbert Simplex Geometry

Frank Nielsen, Ke Sun
Proceedings of 2nd Annual Workshop on Topology, Algebra, and Geometry in Machine Learning (TAG-ML), PMLR 221:254-266, 2023.

Abstract

A key technique of machine learning and computer vision is to embed discrete weighted graphs into continuous spaces for further downstream analysis. Embedding discrete hierarchical structures in hyperbolic geometry has proven very successful since it was shown that any weighted tree can be embedded in that geometry with arbitrary low distortion. Various optimization methods for hyperbolic embeddings based on common models of hyperbolic geometry have been studied. In this paper, we consider Hilbert geometry for the standard simplex which is isometric to a vector space equipped with the variation polytope norm. We study the representation power of this Hilbert simplex geometry by embedding distance matrices of graphs. Our findings demonstrate that Hilbert simplex geometry is competitive to alternative geometries such as the Poincaré hyperbolic ball or the Euclidean geometry for embedding tasks while being fast and numerically robust.

Cite this Paper

BibTeX
@InProceedings{pmlr-v221-nielsen23a, title = {Non-linear Embeddings in Hilbert Simplex Geometry}, author = {Nielsen, Frank and Sun, Ke}, booktitle = {Proceedings of 2nd Annual Workshop on Topology, Algebra, and Geometry in Machine Learning (TAG-ML)}, pages = {254--266}, year = {2023}, editor = {Doster, Timothy and Emerson, Tegan and Kvinge, Henry and Miolane, Nina and Papillon, Mathilde and Rieck, Bastian and Sanborn, Sophia}, volume = {221}, series = {Proceedings of Machine Learning Research}, month = {28 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v221/nielsen23a/nielsen23a.pdf}, url = {https://proceedings.mlr.press/v221/nielsen23a.html}, abstract = {A key technique of machine learning and computer vision is to embed discrete weighted graphs into continuous spaces for further downstream analysis. Embedding discrete hierarchical structures in hyperbolic geometry has proven very successful since it was shown that any weighted tree can be embedded in that geometry with arbitrary low distortion. Various optimization methods for hyperbolic embeddings based on common models of hyperbolic geometry have been studied. In this paper, we consider Hilbert geometry for the standard simplex which is isometric to a vector space equipped with the variation polytope norm. We study the representation power of this Hilbert simplex geometry by embedding distance matrices of graphs. Our findings demonstrate that Hilbert simplex geometry is competitive to alternative geometries such as the Poincaré hyperbolic ball or the Euclidean geometry for embedding tasks while being fast and numerically robust.} }
Endnote
%0 Conference Paper %T Non-linear Embeddings in Hilbert Simplex Geometry %A Frank Nielsen %A Ke Sun %B Proceedings of 2nd Annual Workshop on Topology, Algebra, and Geometry in Machine Learning (TAG-ML) %C Proceedings of Machine Learning Research %D 2023 %E Timothy Doster %E Tegan Emerson %E Henry Kvinge %E Nina Miolane %E Mathilde Papillon %E Bastian Rieck %E Sophia Sanborn %F pmlr-v221-nielsen23a %I PMLR %P 254--266 %U https://proceedings.mlr.press/v221/nielsen23a.html %V 221 %X A key technique of machine learning and computer vision is to embed discrete weighted graphs into continuous spaces for further downstream analysis. Embedding discrete hierarchical structures in hyperbolic geometry has proven very successful since it was shown that any weighted tree can be embedded in that geometry with arbitrary low distortion. Various optimization methods for hyperbolic embeddings based on common models of hyperbolic geometry have been studied. In this paper, we consider Hilbert geometry for the standard simplex which is isometric to a vector space equipped with the variation polytope norm. We study the representation power of this Hilbert simplex geometry by embedding distance matrices of graphs. Our findings demonstrate that Hilbert simplex geometry is competitive to alternative geometries such as the Poincaré hyperbolic ball or the Euclidean geometry for embedding tasks while being fast and numerically robust.
APA
Nielsen, F. & Sun, K.. (2023). Non-linear Embeddings in Hilbert Simplex Geometry. Proceedings of 2nd Annual Workshop on Topology, Algebra, and Geometry in Machine Learning (TAG-ML), in Proceedings of Machine Learning Research 221:254-266 Available from https://proceedings.mlr.press/v221/nielsen23a.html.

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