A Fast and Robust Method for Global Topological Functional Optimization
This addresses the computational inefficiency and instability in topological data analysis for researchers and practitioners, representing an incremental improvement over existing methods.
The paper tackled the problem of optimizing topological functionals using persistence diagrams, which was previously slow, unstable, and fragile. The result was a novel backpropagation scheme that is faster, more stable, and produces more robust optima, with stable visualizations of persistence diagrams.
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.