The decomposition of the higher-order homology embedding constructed from the $k$-Laplacian

• URL: http://arxiv.org/abs/2107.10970v1
• Date: Fri, 23 Jul 2021 00:40:01 GMT
• Title: The decomposition of the higher-order homology embedding constructed from the $k$-Laplacian
• Authors: Yu-Chia Chen, Marina Meil\u{a}
• Abstract summary: The null space of the $k$-th order Laplacian $mathbfmathcal L_k$ encodes the non-trivial topology of a manifold or a network. We propose an algorithm to factorize the homology embedding into subspaces corresponding to a manifold's simplest topological components.
• Score: 5.076419064097734
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The null space of the $k$-th order Laplacian $\mathbf{\mathcal L}_k$, known as the {\em $k$-th homology vector space}, encodes the non-trivial topology of a manifold or a network. Understanding the structure of the homology embedding can thus disclose geometric or topological information from the data. The study of the null space embedding of the graph Laplacian $\mathbf{\mathcal L}_0$ has spurred new research and applications, such as spectral clustering algorithms with theoretical guarantees and estimators of the Stochastic Block Model. In this work, we investigate the geometry of the $k$-th homology embedding and focus on cases reminiscent of spectral clustering. Namely, we analyze the {\em connected sum} of manifolds as a perturbation to the direct sum of their homology embeddings. We propose an algorithm to factorize the homology embedding into subspaces corresponding to a manifold's simplest topological components. The proposed framework is applied to the {\em shortest homologous loop detection} problem, a problem known to be NP-hard in general. Our spectral loop detection algorithm scales better than existing methods and is effective on diverse data such as point clouds and images.

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