Towards Scalable Persistence-Based Topological Optimization

2026-05-09

AI Summary

This work addresses the limitations of persistent homology optimization in high-dimensional settings, which suffers from subsampling bias and sparse gradients. To mitigate these issues, the authors propose a lightweight stochastic slicing subsampling strategy that enhances geometric coverage and introduce a Nadaraya–Watson Gaussian convolution to replace computationally expensive kernel solvers. This approach enables efficient topological gradient smoothing within a reproducing kernel Hilbert space, significantly reducing computational overhead while providing theoretical convergence guarantees. Experimental results demonstrate that the method accelerates the optimization process and achieves superior objective function values compared to existing baselines on both 2D and 3D tasks.

Abstract

Persistence-based topological optimization deforms a point cloud $X \subset \mathbb{R}^d$ by minimizing objectives of the form $L(X) = \ell(\mathrm{Dgm}(X))$, where $\mathrm{Dgm}(X)$ is a persistence diagram. In practice, optimization is limited by two coupled issues: persistent homology is typically computed on subsamples, and the resulting topological gradients are highly sparse, with only a few anchor points receiving nonzero updates. Motivated by diffeomorphic interpolation, which extends sparse gradients to smooth ambient vector fields via Reproducing Kernel Hilbert Space (RKHS) interpolation, we propose a more scalable pipeline that improves both subsampling and gradient extension. We introduce subsampling via random slicing, a lightweight scheme that promotes iteration-wise geometric coverage and mitigates density bias. We further replace the costly kernel solve with a fast Nadaraya-Watson (NW) Gaussian convolution, producing a globally defined smooth update field at a fraction of the computational cost, while being more suited for topological optimization tasks. We provide theoretical guarantees for NW smoothing, including anchor approximation bounds and global Lipschitz estimates. Experiments in $2$D and $3$D show that combining random slicing with NW smoothing yields consistent speedups and improved objective values over other baselines on common persistence losses.

Problem

• persistence-based optimization
• sparse gradients
• subsampling
• topological optimization
• persistent homology

Innovation

• random slicing
• Nadaraya-Watson smoothing
• topological optimization
• persistence diagram
• scalable gradient extension