A Fast and Robust Method for Global Topological Functional Optimization

Paul Bendich, Yitzchak Solomon, Alexander Wagner

Abstract:

Topological statistics, in the form of persis-
tence diagrams, are a class of shape descrip-
tors that capture global structural informa-
tion in data. The mapping from data struc-
tures to persistence diagrams is almost every-
where differentiable, allowing for topological
gradients to be backpropagated to ordinary
gradients. However, as a method for optimiz-
ing a topological functional, this backprop-
agation method is expensive, unstable, and
produces very fragile optima. Our contribu-
tion is to introduce a novel backpropagation
scheme that is significantly faster, more sta-
ble, and produces more robust optima. More-
over, 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.