Differentiability and Optimization of Multiparameter Persistent Homology

TL;DR

This paper develops a general framework for differentiability and optimization of multiparameter persistent homology descriptors, enabling improved data analysis and optimization in complex geometric data settings.

Contribution

It introduces a unified framework for differentiability of various multiparameter homological descriptors, broadening the scope of topological optimization methods.

Findings

Optimizing multiparameter descriptors can outperform one-parameter methods.

The framework includes well-known descriptors like signed barcodes and persistence landscapes.

Numerical experiments demonstrate practical benefits of the proposed approach.

Abstract

Real-valued functions on geometric data -- such as node attributes on a graph -- can be optimized using descriptors from persistent homology, allowing the user to incorporate topological terms in the loss function. When optimizing a single real-valued function (the one-parameter setting), there is a canonical choice of descriptor for persistent homology: the barcode. The operation mapping a real-valued function to its barcode is differentiable almost everywhere, and the convergence of gradient descent for losses using barcodes is relatively well understood. When optimizing a vector-valued function (the multiparameter setting), there is no unique choice of descriptor for multiparameter persistent homology, and many distinct descriptors have been proposed. This calls for the development of a general framework for differentiability and optimization that applies to a wide range of multiparameter homological descriptors. In this article, we develop such a framework and show that it encompasses well-known descriptors of different flavors, such as signed barcodes and the multiparameter persistence landscape. We complement the theory with numerical experiments supporting the idea that optimizing multiparameter homological descriptors can lead to improved performances compared to optimizing one-parameter descriptors, even when using the simplest and most efficiently computable multiparameter descriptors.