On the Convergence of Optimizing Persistent-Homology-Based Losses

TL;DR

This paper introduces a new regularized topological loss function based on persistent homology, which guarantees efficient optimization and convergence, addressing the oscillation issues in previous topological losses.

Contribution

It proposes a novel regularization term and modifies existing topological loss to ensure better convergence and theoretical guarantees during optimization.

Findings

The new loss enforces desired topological properties effectively.

The modified loss achieves guaranteed convergence under mild assumptions.

The approach improves optimization stability in topological loss applications.

Abstract

Topological loss based on persistent homology has shown promise in various applications. A topological loss enforces the model to achieve certain desired topological property. Despite its empirical success, less is known about the optimization behavior of the loss. In fact, the topological loss involves combinatorial configurations that may oscillate during optimization. In this paper, we introduce a general purpose regularized topology-aware loss. We propose a novel regularization term and also modify existing topological loss. These contributions lead to a new loss function that not only enforces the model to have desired topological behavior, but also achieves satisfying convergence behavior. Our main theoretical result guarantees that the loss can be optimized efficiently, under mild assumptions.