A Fast and Robust Method for Global Topological Functional Optimization
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
@article{arxiv.2009.08496,
title = {A Fast and Robust Method for Global Topological Functional Optimization},
author = {Elchanan Solomon and Alexander Wagner and Paul Bendich},
journal= {arXiv preprint arXiv:2009.08496},
year = {2021}
}
Comments
Added new experiments: one on robustness, the other a cell segmentation task. Other parts of the paper were clarified by including more background exposition