Harmonic Decompositions of Convolutional Networks

Meyer Scetbon, Zaid Harchaoui
Proceedings of the 37th International Conference on Machine Learning, PMLR 119:8522-8532, 2020.

Abstract

We present a description of the function space and the smoothness class associated with a convolutional network using the machinery of reproducing kernel Hilbert spaces. We show that the mapping associated with a convolutional network expands into a sum involving elementary functions akin to spherical harmonics. This functional decomposition can be related to the functional ANOVA decomposition in nonparametric statistics. Building off our functional characterization of convolutional networks, we obtain statistical bounds highlighting an interesting trade-off between the approximation error and the estimation error.

Cite this Paper


BibTeX
@InProceedings{pmlr-v119-scetbon20a, title = {Harmonic Decompositions of Convolutional Networks}, author = {Scetbon, Meyer and Harchaoui, Zaid}, booktitle = {Proceedings of the 37th International Conference on Machine Learning}, pages = {8522--8532}, year = {2020}, editor = {III, Hal Daumé and Singh, Aarti}, volume = {119}, series = {Proceedings of Machine Learning Research}, month = {13--18 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v119/scetbon20a/scetbon20a.pdf}, url = {http://proceedings.mlr.press/v119/scetbon20a.html}, abstract = {We present a description of the function space and the smoothness class associated with a convolutional network using the machinery of reproducing kernel Hilbert spaces. We show that the mapping associated with a convolutional network expands into a sum involving elementary functions akin to spherical harmonics. This functional decomposition can be related to the functional ANOVA decomposition in nonparametric statistics. Building off our functional characterization of convolutional networks, we obtain statistical bounds highlighting an interesting trade-off between the approximation error and the estimation error.} }
Endnote
%0 Conference Paper %T Harmonic Decompositions of Convolutional Networks %A Meyer Scetbon %A Zaid Harchaoui %B Proceedings of the 37th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2020 %E Hal Daumé III %E Aarti Singh %F pmlr-v119-scetbon20a %I PMLR %P 8522--8532 %U http://proceedings.mlr.press/v119/scetbon20a.html %V 119 %X We present a description of the function space and the smoothness class associated with a convolutional network using the machinery of reproducing kernel Hilbert spaces. We show that the mapping associated with a convolutional network expands into a sum involving elementary functions akin to spherical harmonics. This functional decomposition can be related to the functional ANOVA decomposition in nonparametric statistics. Building off our functional characterization of convolutional networks, we obtain statistical bounds highlighting an interesting trade-off between the approximation error and the estimation error.
APA
Scetbon, M. & Harchaoui, Z.. (2020). Harmonic Decompositions of Convolutional Networks. Proceedings of the 37th International Conference on Machine Learning, in Proceedings of Machine Learning Research 119:8522-8532 Available from http://proceedings.mlr.press/v119/scetbon20a.html.

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