Published in Proceedings of the 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 $K\hookrightarrow L$ 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 $f: K \to L$. 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 $K\subset L$. However, the simplicial map setting is much more challenging than the inclusion setting since the underlying algebraic structure is much more complicated. Read more
Recommended citation: Gülen, A.B., Mémoli, F., Wan, Z., Wang, Y. (2023). A Generalization of the Persistent Laplacian to Simplicial Maps. 39th International Symposium on Computational Geometry (SoCG 2023).
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