While persistent homology has taken strides towards becoming a widespread tool for data analysis, multidimensional persistence has proven more difficult to apply. One reason is the serious drawback of no longer having a concise and complete descriptor analogous to the persistence diagrams of the former. We propose a simple algebraic construction to illustrate the existence of infinite families of indecomposable persistence modules over regular grids of sufficient size. On top of providing a constructive proof of representation infinite type, we also provide realizations by topological spaces and Vietoris-Rips filtrations, showing that they can actually appear in real data and are not the product of degeneracies.
@InProceedings{buchet_et_al:LIPIcs.SoCG.2018.15, author = {Buchet, Micka\"{e}l and Escolar, Emerson G.}, title = {{Realizations of Indecomposable Persistence Modules of Arbitrarily Large Dimension}}, booktitle = {34th International Symposium on Computational Geometry (SoCG 2018)}, pages = {15:1--15:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-066-8}, ISSN = {1868-8969}, year = {2018}, volume = {99}, editor = {Speckmann, Bettina and T\'{o}th, Csaba D.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.15}, URN = {urn:nbn:de:0030-drops-87287}, doi = {10.4230/LIPIcs.SoCG.2018.15}, annote = {Keywords: persistent homology, multi-persistence, representation theory, quivers, commutative ladders, Vietoris-Rips filtration} }
Feedback for Dagstuhl Publishing