A Fast and Robust Method for Global Topological Functional Optimization

Yitzchak Solomon, Alexander Wagner, Paul Bendich
Proceedings of The 24th International Conference on Artificial Intelligence and Statistics, PMLR 130:109-117, 2021.

Abstract

Topological statistics, in the form of persistence diagrams, are a class of shape descriptors that capture global structural information in data. The mapping from data structures to persistence diagrams is almost everywhere differentiable, allowing for topological gradients to be backpropagated to ordinary gradients. However, as a method for optimizing a topological functional, this backpropagation method is expensive, unstable, and produces very fragile optima. Our contribution is to introduce a novel backpropagation scheme that is significantly faster, more stable, and produces more robust optima. Moreover, this scheme can also be used to produce a stable visualization of dots in a persistence diagram as a distribution over critical, and near-critical, simplices in the data structure.

Cite this Paper

BibTeX
@InProceedings{pmlr-v130-solomon21a, title = { A Fast and Robust Method for Global Topological Functional Optimization }, author = {Solomon, Yitzchak and Wagner, Alexander and Bendich, Paul}, booktitle = {Proceedings of The 24th International Conference on Artificial Intelligence and Statistics}, pages = {109--117}, 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/solomon21a/solomon21a.pdf}, url = {https://proceedings.mlr.press/v130/solomon21a.html}, abstract = { Topological statistics, in the form of persistence diagrams, are a class of shape descriptors that capture global structural information in data. The mapping from data structures to persistence diagrams is almost everywhere differentiable, allowing for topological gradients to be backpropagated to ordinary gradients. However, as a method for optimizing a topological functional, this backpropagation method is expensive, unstable, and produces very fragile optima. Our contribution is to introduce a novel backpropagation scheme that is significantly faster, more stable, and produces more robust optima. Moreover, this scheme can also be used to produce a stable visualization of dots in a persistence diagram as a distribution over critical, and near-critical, simplices in the data structure. } }
Endnote
%0 Conference Paper %T A Fast and Robust Method for Global Topological Functional Optimization %A Yitzchak Solomon %A Alexander Wagner %A Paul Bendich %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-solomon21a %I PMLR %P 109--117 %U https://proceedings.mlr.press/v130/solomon21a.html %V 130 %X Topological statistics, in the form of persistence diagrams, are a class of shape descriptors that capture global structural information in data. The mapping from data structures to persistence diagrams is almost everywhere differentiable, allowing for topological gradients to be backpropagated to ordinary gradients. However, as a method for optimizing a topological functional, this backpropagation method is expensive, unstable, and produces very fragile optima. Our contribution is to introduce a novel backpropagation scheme that is significantly faster, more stable, and produces more robust optima. Moreover, this scheme can also be used to produce a stable visualization of dots in a persistence diagram as a distribution over critical, and near-critical, simplices in the data structure.
APA
Solomon, Y., Wagner, A. & Bendich, P.. (2021). A Fast and Robust Method for Global Topological Functional Optimization . Proceedings of The 24th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 130:109-117 Available from https://proceedings.mlr.press/v130/solomon21a.html.

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