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Abstract A central challenge in topological data analysis is the interpretation of barcodes. The classical algebraic-topological approach to interpreting homology classes is to build maps to spaces whose homology carries semantics we understand and then to appeal to functoriality. However, we often lack such maps in real data; instead, we must rely on a cross-dissimilarity measure between our observations of a system and a reference. In this paper, we develop a pair of computational homological algebra approaches for relating persistent homology classes and barcodes:persistent extension, which enumerates potential relations between homology classes from two complexes built on the same vertex set, and the method ofanalogous bars, which utilizes persistent extension and the witness complex built from a cross-dissimilarity measure to provide relations across systems. We provide an implementation of these methods and demonstrate their use in comparing homology classes between two samples from the same metric space and determining whether topology is maintained or destroyed under clustering and dimensionality reduction.
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This content will become publicly available on January 1, 2026
Persistent Topological Laplacians—A Survey
Persistent topological Laplacians constitute a new class of tools in topological data analysis (TDA). They are motivated by the necessity to address challenges encountered in persistent homology when handling complex data. These Laplacians combine multiscale analysis with topological techniques to characterize the topological and geometrical features of functions and data. Their kernels fully retrieve the topological invariants of corresponding persistent homology, while their non-harmonic spectra provide supplementary information. Persistent topological Laplacians have demonstrated superior performance over persistent homology in the analysis of large-scale protein engineering datasets. In this survey, we offer a pedagogical review of persistent topological Laplacians formulated in various mathematical settings, including simplicial complexes, path complexes, flag complexes, digraphs, hypergraphs, hyperdigraphs, cellular sheaves, and N-chain complexes.
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- Award ID(s):
- 2052983
- PAR ID:
- 10616122
- Publisher / Repository:
- Mathematics
- Date Published:
- Journal Name:
- Mathematics
- Volume:
- 13
- Issue:
- 2
- ISSN:
- 2227-7390
- Page Range / eLocation ID:
- 208
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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