Product Manifold Learning

Sharon Zhang, Amit Moscovich, Amit Singer
Proceedings of The 24th International Conference on Artificial Intelligence and Statistics, PMLR 130:3241-3249, 2021.

Abstract

We consider dimensionality reduction for data sets with two or more independent degrees of freedom. For example, measurements of deformable shapes with several parts that move independently fall under this characterization. Mathematically, if the space of each continuous independent motion is a manifold, then their combination forms a product manifold. In this paper, we present an algorithm for manifold factorization given a sample of points from the product manifold. Our algorithm is based on spectral graph methods for manifold learning and the separability of the Laplacian operator on product spaces. Recovering the factors of a manifold yields meaningful lower-dimensional representations, allowing one to focus on particular aspects of the data space while ignoring others. We demonstrate the potential use of our method for an important and challenging problem in structural biology: mapping the motions of proteins and other large molecules using cryo-electron microscopy data sets.

Cite this Paper

BibTeX
@InProceedings{pmlr-v130-zhang21j, title = { Product Manifold Learning }, author = {Zhang, Sharon and Moscovich, Amit and Singer, Amit}, booktitle = {Proceedings of The 24th International Conference on Artificial Intelligence and Statistics}, pages = {3241--3249}, year = {2021}, editor = {Banerjee, Arindam and Fukumizu, Kenji}, volume = {130}, series = {Proceedings of Machine Learning Research}, month = {13--15 Apr}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v130/zhang21j/zhang21j.pdf}, url = {https://proceedings.mlr.press/v130/zhang21j.html}, abstract = { We consider dimensionality reduction for data sets with two or more independent degrees of freedom. For example, measurements of deformable shapes with several parts that move independently fall under this characterization. Mathematically, if the space of each continuous independent motion is a manifold, then their combination forms a product manifold. In this paper, we present an algorithm for manifold factorization given a sample of points from the product manifold. Our algorithm is based on spectral graph methods for manifold learning and the separability of the Laplacian operator on product spaces. Recovering the factors of a manifold yields meaningful lower-dimensional representations, allowing one to focus on particular aspects of the data space while ignoring others. We demonstrate the potential use of our method for an important and challenging problem in structural biology: mapping the motions of proteins and other large molecules using cryo-electron microscopy data sets. } }
Endnote
%0 Conference Paper %T Product Manifold Learning %A Sharon Zhang %A Amit Moscovich %A Amit Singer %B Proceedings of The 24th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2021 %E Arindam Banerjee %E Kenji Fukumizu %F pmlr-v130-zhang21j %I PMLR %P 3241--3249 %U https://proceedings.mlr.press/v130/zhang21j.html %V 130 %X We consider dimensionality reduction for data sets with two or more independent degrees of freedom. For example, measurements of deformable shapes with several parts that move independently fall under this characterization. Mathematically, if the space of each continuous independent motion is a manifold, then their combination forms a product manifold. In this paper, we present an algorithm for manifold factorization given a sample of points from the product manifold. Our algorithm is based on spectral graph methods for manifold learning and the separability of the Laplacian operator on product spaces. Recovering the factors of a manifold yields meaningful lower-dimensional representations, allowing one to focus on particular aspects of the data space while ignoring others. We demonstrate the potential use of our method for an important and challenging problem in structural biology: mapping the motions of proteins and other large molecules using cryo-electron microscopy data sets.
APA
Zhang, S., Moscovich, A. & Singer, A.. (2021). Product Manifold Learning . Proceedings of The 24th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 130:3241-3249 Available from https://proceedings.mlr.press/v130/zhang21j.html.

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