Regularization of Persistent Homology Gradient Computation
This addresses a bottleneck for researchers in topological data analysis and deep learning who need differentiable persistent homology for integration into neural networks, representing an incremental improvement.
The paper tackles the problem of computing gradients for persistent homology, which is necessary for integrating it into neural networks but is ill-posed with infinitely many solutions, by proposing a novel regularization method using a grouping term to ensure gradients are defined with respect to larger entities rather than individual points.
Persistent homology is a method for computing the topological features present in a given data. Recently, there has been much interest in the integration of persistent homology as a computational step in neural networks or deep learning. In order for a given computation to be integrated in such a way, the computation in question must be differentiable. Computing the gradients of persistent homology is an ill-posed inverse problem with infinitely many solutions. Consequently, it is important to perform regularization so that the solution obtained agrees with known priors. In this work we propose a novel method for regularizing persistent homology gradient computation through the addition of a grouping term. This has the effect of helping to ensure gradients are defined with respect to larger entities and not individual points.