🤖 AI Summary
This work addresses the challenge of directly controlling Betti numbers and persistent homology in optimization or generation tasks due to their discrete and combinatorial nature. It proposes the first framework that integrates Hodge spectral theory with differentiable optimization, constructing a spectral relaxation of the Hodge–Laplacian via soft graphs and soft clique complexes. By introducing spectral proxies for zero and near-zero modes, the method enables smooth, geometry-aware gradient updates toward topological objectives. The approach unifies treatment of Vietoris–Rips filtrations and graph clique complexes, yielding more uniform spatial gradients and smoother scale normalization in point cloud tasks, while effectively regulating normalized first Betti numbers on graphs and supporting joint optimization with standard graph features.
📝 Abstract
Topological information is widely used in data analysis, network design, and machine learning, and topological constraints naturally arise when optimizing or generating objects with prescribed homological structure. However, directly controlling Betti numbers and persistent homology is difficult because they are discrete and combinatorial. We propose a differentiable framework for topology-constrained optimization based on Hodge-spectral relaxations of homological constraints and low-pass spectral filters. From soft graphs and soft clique complexes, we construct Hodge-Laplacian-type spectral relaxations that unify graph clique complexes and Vietoris--Rips filtrations of point clouds. In the hard limit, the penalty-regularized ambient operator recovers the ordinary Hodge Laplacian on the active subcomplex, while in the soft regime it serves as a differentiable low-frequency spectral surrogate. Homological information is represented by zero and near-zero modes, and differentiable topological objectives are defined using heat filters, resolvent filters, and polynomial Laplacian moments. For point clouds, we show that the proposed Hodge spectral-filter losses yield more spatially distributed gradients, smoother scale-normalized behavior under persistence-pairing changes, and geometry-aware update directions than persistent-homology-based losses. For graph clique complexes, Laplacian moments control normalized first-Betti-type quantities and can be combined with ordinary graph-feature objectives. We also discuss connections to trace-based normalized Betti-number estimation, polynomial spectral methods, and possible quantum trace estimation.