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

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

23 Jul 2021NeurIPS 2021 12arXiv:2107.10970archive 2025-07-28

Yu-Chia Chen, Marina Meilă

The null space of the k-th order Laplacian ℒₖ, 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 ℒ₀ 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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