Geometric Wavelet Scattering Networks on Compact Riemannian Manifolds

Michael Perlmutter, Feng Gao, Guy Wolf, Matthew Hirn
Proceedings of The First Mathematical and Scientific Machine Learning Conference, PMLR 107:570-604, 2020.

Abstract

The Euclidean scattering transform was introduced nearly a decade ago to improve the mathematical understanding of convolutional neural networks. Inspired by recent interest in geometric deep learning, which aims to generalize convolutional neural networks to manifold and graph-structured domains, we define a geometric scattering transform on manifolds. Similar to the Euclidean scattering transform, the geometric scattering transform is based on a cascade of wavelet filters and pointwise nonlinearities. It is invariant to local isometries and stable to certain types of diffeomorphisms. Empirical results demonstrate its utility on several geometric learning tasks. Our results generalize the deformation stability and local translation invariance of Euclidean scattering, and demonstrate the importance of linking the used filter structures to the underlying geometry of the data.

Cite this Paper

BibTeX
@InProceedings{pmlr-v107-perlmutter20a, title = {Geometric Wavelet Scattering Networks on Compact {R}iemannian Manifolds}, author = {Perlmutter, Michael and Gao, Feng and Wolf, Guy and Hirn, Matthew}, booktitle = {Proceedings of The First Mathematical and Scientific Machine Learning Conference}, pages = {570--604}, year = {2020}, editor = {Lu, Jianfeng and Ward, Rachel}, volume = {107}, series = {Proceedings of Machine Learning Research}, month = {20--24 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v107/perlmutter20a/perlmutter20a.pdf}, url = {https://proceedings.mlr.press/v107/perlmutter20a.html}, abstract = { The Euclidean scattering transform was introduced nearly a decade ago to improve the mathematical understanding of convolutional neural networks. Inspired by recent interest in geometric deep learning, which aims to generalize convolutional neural networks to manifold and graph-structured domains, we define a geometric scattering transform on manifolds. Similar to the Euclidean scattering transform, the geometric scattering transform is based on a cascade of wavelet filters and pointwise nonlinearities. It is invariant to local isometries and stable to certain types of diffeomorphisms. Empirical results demonstrate its utility on several geometric learning tasks. Our results generalize the deformation stability and local translation invariance of Euclidean scattering, and demonstrate the importance of linking the used filter structures to the underlying geometry of the data. } }
Endnote
%0 Conference Paper %T Geometric Wavelet Scattering Networks on Compact Riemannian Manifolds %A Michael Perlmutter %A Feng Gao %A Guy Wolf %A Matthew Hirn %B Proceedings of The First Mathematical and Scientific Machine Learning Conference %C Proceedings of Machine Learning Research %D 2020 %E Jianfeng Lu %E Rachel Ward %F pmlr-v107-perlmutter20a %I PMLR %P 570--604 %U https://proceedings.mlr.press/v107/perlmutter20a.html %V 107 %X The Euclidean scattering transform was introduced nearly a decade ago to improve the mathematical understanding of convolutional neural networks. Inspired by recent interest in geometric deep learning, which aims to generalize convolutional neural networks to manifold and graph-structured domains, we define a geometric scattering transform on manifolds. Similar to the Euclidean scattering transform, the geometric scattering transform is based on a cascade of wavelet filters and pointwise nonlinearities. It is invariant to local isometries and stable to certain types of diffeomorphisms. Empirical results demonstrate its utility on several geometric learning tasks. Our results generalize the deformation stability and local translation invariance of Euclidean scattering, and demonstrate the importance of linking the used filter structures to the underlying geometry of the data.
APA
Perlmutter, M., Gao, F., Wolf, G. & Hirn, M.. (2020). Geometric Wavelet Scattering Networks on Compact Riemannian Manifolds. Proceedings of The First Mathematical and Scientific Machine Learning Conference, in Proceedings of Machine Learning Research 107:570-604 Available from https://proceedings.mlr.press/v107/perlmutter20a.html.

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