TY - GEN
T1 - Decomposition of Zero-Dimensional Persistence Modules via Rooted Subsets
AU - Alonso, Ángel Javier
AU - Kerber, Michael
N1 - Publisher Copyright:
© Ángel Javier Alonso and Michael Kerber; licensed under Creative Commons License CC-BY 4.0.
PY - 2023/6/1
Y1 - 2023/6/1
N2 - We study the decomposition of zero-dimensional persistence modules, viewed as functors valued in the category of vector spaces factorizing through sets. Instead of working directly at the level of vector spaces, we take a step back and first study the decomposition problem at the level of sets. This approach allows us to define the combinatorial notion of rooted subsets. In the case of a filtered metric space M, rooted subsets relate the clustering behavior of the points of M with the decomposition of the associated persistence module. In particular, we can identify intervals in such a decomposition quickly. In addition, rooted subsets can be understood as a generalization of the elder rule, and are also related to the notion of constant conqueror of Cai, Kim, Mémoli and Wang. As an application, we give a lower bound on the number of intervals that we can expect in the decomposition of zero-dimensional persistence modules of a density-Rips filtration in Euclidean space: in the limit, and under very general circumstances, we can expect that at least 25% of the indecomposable summands are interval modules.
AB - We study the decomposition of zero-dimensional persistence modules, viewed as functors valued in the category of vector spaces factorizing through sets. Instead of working directly at the level of vector spaces, we take a step back and first study the decomposition problem at the level of sets. This approach allows us to define the combinatorial notion of rooted subsets. In the case of a filtered metric space M, rooted subsets relate the clustering behavior of the points of M with the decomposition of the associated persistence module. In particular, we can identify intervals in such a decomposition quickly. In addition, rooted subsets can be understood as a generalization of the elder rule, and are also related to the notion of constant conqueror of Cai, Kim, Mémoli and Wang. As an application, we give a lower bound on the number of intervals that we can expect in the decomposition of zero-dimensional persistence modules of a density-Rips filtration in Euclidean space: in the limit, and under very general circumstances, we can expect that at least 25% of the indecomposable summands are interval modules.
KW - Clustering
KW - Decomposition of persistence modules
KW - Elder Rule
KW - Multiparameter persistent homology
UR - http://www.scopus.com/inward/record.url?scp=85163427629&partnerID=8YFLogxK
U2 - 10.4230/LIPIcs.SoCG.2023.7
DO - 10.4230/LIPIcs.SoCG.2023.7
M3 - Conference paper
AN - SCOPUS:85163427629
T3 - Leibniz International Proceedings in Informatics, LIPIcs
BT - 39th International Symposium on Computational Geometry, SoCG 2023
A2 - Chambers, Erin W.
A2 - Gudmundsson, Joachim
PB - Schloss Dagstuhl - Leibniz-Zentrum für Informatik
T2 - 39th International Symposium on Computational Geometry
Y2 - 12 June 2023 through 15 June 2023
ER -