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A Generalization of the Persistent Laplacian to Simplicial Maps

Authors: Aziz Burak Gülen, Facundo Mémoli, Zhengchao Wan, and Yusu Wang

Published in: LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023)

The (combinatorial) graph Laplacian is a fundamental object in the analysis of, and optimization on, graphs. Via a topological view, this operator can be extended to a simplicial complex K and therefore offers a way to perform "signal processing" on p-(co)chains of K. Recently, the concept of persistent Laplacian was proposed and studied for a pair of simplicial complexes K ↪ L connected by an inclusion relation, further broadening the use of Laplace-based operators. In this paper, we significantly expand the scope of the persistent Laplacian by generalizing it to a pair of weighted simplicial complexes connected by a weight preserving simplicial map f: K → L. Such a simplicial map setting arises frequently, e.g., when relating a coarsened simplicial representation with an original representation, or the case when the two simplicial complexes are spanned by different point sets, i.e. cases in which it does not hold that K ⊂ L. However, the simplicial map setting is much more challenging than the inclusion setting since the underlying algebraic structure is much more complicated. We present a natural generalization of the persistent Laplacian to the simplicial setting. To shed insight on the structure behind it, as well as to develop an algorithm to compute it, we exploit the relationship between the persistent Laplacian and the Schur complement of a matrix. A critical step is to view the Schur complement as a functorial way of restricting a self-adjoint positive semi-definite operator to a given subspace. As a consequence of this relation, we prove that the qth persistent Betti number of the simplicial map f: K → L equals the nullity of the qth persistent Laplacian Δ_q^{K,L}. We then propose an algorithm for finding the matrix representation of Δ_q^{K,L} which in turn yields a fundamentally different algorithm for computing the qth persistent Betti number of a simplicial map. Finally, we study the persistent Laplacian on simplicial towers under weight-preserving simplicial maps and establish monotonicity results for their eigenvalues.

Cite as

Aziz Burak Gülen, Facundo Mémoli, Zhengchao Wan, and Yusu Wang. A Generalization of the Persistent Laplacian to Simplicial Maps. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 37:1-37:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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  author =	{G\"{u}len, Aziz Burak and M\'{e}moli, Facundo and Wan, Zhengchao and Wang, Yusu},
  title =	{{A Generalization of the Persistent Laplacian to Simplicial Maps}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{37:1--37:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-273-0},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{258},
  editor =	{Chambers, Erin W. and Gudmundsson, Joachim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.37},
  URN =		{urn:nbn:de:0030-drops-178877},
  doi =		{10.4230/LIPIcs.SoCG.2023.37},
  annote =	{Keywords: combinatorial Laplacian, persistent Laplacian, Schur complement, persistent homology, persistent Betti number}
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