Year: 2024
Type: preprint
Abstract: We propose a modification of the Mapper algorithm that removes the assumption of a single resolution scale across semantic space and improves the robustness of the results under change of parameters. Our work is motivated by datasets where the density in the image of the Morse-type function (the lens-space density) varies widely. For such datasets, tuning the resolution parameter of Mapper is difficult because small changes can lead to significant variations in the output. By improving the robustness of the output under these variations, our method makes it easier to tune the resolution for datasets with highly variable lens-space density. This improvement is achieved by generalising the type of permitted cover for Mapper and incorporating the lens-space density into the cover. Furthermore, we prove that for covers satisfying natural assumptions, the graph produced by Mapper still converges in bottleneck distance to the Reeb graph of the Rips complex of the data, while possibly capturing more topological features than a standard Mapper cover. Finally, we discuss implementation details and present the results of computational experiments. We also provide an accompanying reference implementation.
Source: arXiv (Cornell University)
Authors: Ruscitti, Kaleb D., McInnes, Leland
Institutions: University of Waterloo, Government of Canada
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
Funder: Government of Canada
OpenAlex ID: W4403322200