HoroPCA: Hyperbolic Dimensionality Reduction via Horospherical Projections

Ines Chami, Albert Gu, Dat P Nguyen, Christopher Re
Proceedings of the 38th International Conference on Machine Learning, PMLR 139:1419-1429, 2021.

Abstract

This paper studies Principal Component Analysis (PCA) for data lying in hyperbolic spaces. Given directions, PCA relies on: (1) a parameterization of subspaces spanned by these directions, (2) a method of projection onto subspaces that preserves information in these directions, and (3) an objective to optimize, namely the variance explained by projections. We generalize each of these concepts to the hyperbolic space and propose HoroPCA, a method for hyperbolic dimensionality reduction. By focusing on the core problem of extracting principal directions, HoroPCA theoretically better preserves information in the original data such as distances, compared to previous generalizations of PCA. Empirically, we validate that HoroPCA outperforms existing dimensionality reduction methods, significantly reducing error in distance preservation. As a data whitening method, it improves downstream classification by up to 3.9% compared to methods that don’t use whitening. Finally, we show that HoroPCA can be used to visualize hyperbolic data in two dimensions.

Cite this Paper

BibTeX
@InProceedings{pmlr-v139-chami21a, title = {HoroPCA: Hyperbolic Dimensionality Reduction via Horospherical Projections}, author = {Chami, Ines and Gu, Albert and Nguyen, Dat P and Re, Christopher}, booktitle = {Proceedings of the 38th International Conference on Machine Learning}, pages = {1419--1429}, year = {2021}, editor = {Meila, Marina and Zhang, Tong}, volume = {139}, series = {Proceedings of Machine Learning Research}, month = {18--24 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v139/chami21a/chami21a.pdf}, url = {https://proceedings.mlr.press/v139/chami21a.html}, abstract = {This paper studies Principal Component Analysis (PCA) for data lying in hyperbolic spaces. Given directions, PCA relies on: (1) a parameterization of subspaces spanned by these directions, (2) a method of projection onto subspaces that preserves information in these directions, and (3) an objective to optimize, namely the variance explained by projections. We generalize each of these concepts to the hyperbolic space and propose HoroPCA, a method for hyperbolic dimensionality reduction. By focusing on the core problem of extracting principal directions, HoroPCA theoretically better preserves information in the original data such as distances, compared to previous generalizations of PCA. Empirically, we validate that HoroPCA outperforms existing dimensionality reduction methods, significantly reducing error in distance preservation. As a data whitening method, it improves downstream classification by up to 3.9% compared to methods that don’t use whitening. Finally, we show that HoroPCA can be used to visualize hyperbolic data in two dimensions.} }
Endnote
%0 Conference Paper %T HoroPCA: Hyperbolic Dimensionality Reduction via Horospherical Projections %A Ines Chami %A Albert Gu %A Dat P Nguyen %A Christopher Re %B Proceedings of the 38th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2021 %E Marina Meila %E Tong Zhang %F pmlr-v139-chami21a %I PMLR %P 1419--1429 %U https://proceedings.mlr.press/v139/chami21a.html %V 139 %X This paper studies Principal Component Analysis (PCA) for data lying in hyperbolic spaces. Given directions, PCA relies on: (1) a parameterization of subspaces spanned by these directions, (2) a method of projection onto subspaces that preserves information in these directions, and (3) an objective to optimize, namely the variance explained by projections. We generalize each of these concepts to the hyperbolic space and propose HoroPCA, a method for hyperbolic dimensionality reduction. By focusing on the core problem of extracting principal directions, HoroPCA theoretically better preserves information in the original data such as distances, compared to previous generalizations of PCA. Empirically, we validate that HoroPCA outperforms existing dimensionality reduction methods, significantly reducing error in distance preservation. As a data whitening method, it improves downstream classification by up to 3.9% compared to methods that don’t use whitening. Finally, we show that HoroPCA can be used to visualize hyperbolic data in two dimensions.
APA
Chami, I., Gu, A., Nguyen, D.P. & Re, C.. (2021). HoroPCA: Hyperbolic Dimensionality Reduction via Horospherical Projections. Proceedings of the 38th International Conference on Machine Learning, in Proceedings of Machine Learning Research 139:1419-1429 Available from https://proceedings.mlr.press/v139/chami21a.html.

Related Material