Hodge Spectral Surrogates for Topology-Constrained Optimization

[2606.25194] Hodge Spectral Surrogates for Topology-Constrained Optimization Skip to main content Search Submit Donate Log in Mathematics > Algebraic Topology arXiv:2606.25194 (math) [Submitted on 23 Jun 2026 ( v1 ), last revised 25 Aug 2026 (this version, v2)] Title: Hodge Spectral Surrogates for Topology-Constrained Optimization Authors: Satoshi Kanno , Yoshi-aki Shimada View PDF HTML (experimental) Abstract: Topology-constrained optimization arises in applications such as medical image segmentation, geometric design, and network generation, where global structural correctness matters. Yet Betti numbers and persistent homology are discrete and combinatorial, making them difficult to control directly by gradient-based optimization. We propose a differentiable framework that replaces discrete topological counting with a soft spectral relaxation of the Hodge Laplacian. Candidate simplices are embedded in a fixed ambient complex, while simplex inclusion is represented by continuous activations, producing a smooth spectral path between different hard complexes. The soft states are used only as optimization surrogates; in the hard limit, the ordinary Hodge Laplacian and its homological zero modes are recovered. We establish operator-norm and spectral perturbation bounds connecting hard Betti numbers to soft near-zero modes, and derive differentiable objectives using heat, resolvent, and polynomial low-pass filters. Controlled experiments on Vietoris--Rips point clouds show more spatially distributed gradients, reduced scale-normalized derivative variation near persistence-pairing changes, and geometry-aware update directions in the tested settings. Experiments on graph clique complexes further show controllable shifts in sampled hard Betti regimes and compatibility with standard graph objectives. The framework provides a complementary Hodge-spectral approach to persistent-homology-based topology optimization. Comments: 57 pages, 11 figures Subjects: Algebraic Topology (math.AT) ; Computational Geometry (cs.CG); Mathematical Physics (math-ph); Quantum Physics (quant-ph) Cite as: arXiv:2606.25194 [math.AT] (or arXiv:2606.25194v2 [math.AT] for this version) https://doi.org/10.48550/arXiv.2606.25194 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Satoshi Kanno [ view email ] [v1] Tue, 23 Jun 2026 21:38:32 UTC (832 KB) [v2] Tue, 25 Aug 2026 06:01:36 UTC (1,375 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.AT < prev | next > new | recent | 2026-06 Change to browse by: cs cs.CG math math-ph math.MP quant-ph References & Citations INSPIRE HEP SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

Related documents

• Hodge Spectral Surrogates for Topology-Constrained Optimization#670652
• Optimization Geometry of Equivalent Brownian RKHS Representations#1036084
• Optimization Geometry of Equivalent Brownian RKHS Representations#1006874
• High-Probability Convergence Theory for Distributed Composite Optimization with Sub-Weibull Noises#160903
• MF-SCBO : Multi-fidelity Scalable Constrained Bayesian Optimization#1078012