TY - GEN
T1 - A Persistent Version of Latschev’s Theorem
AU - Oudot, Steve
AU - Waas, Lukas
N1 - Publisher Copyright:
© Steve Oudot and Lukas Waas;
PY - 2026/5/27
Y1 - 2026/5/27
N2 - Latschev’s theorem provides sufficient conditions on a metric space M and δ > 0 for the homotopy type of M to agree with that of the Vietoris-Rips complex Rδ(M) of any nearby space M in the Gromov-Hausdorff distance. We prove a persistent version of this theorem, providing sufficient conditions on a pair (M, f : M → RN) and δ > 0 for the persistent homotopy type of the sublevel set filtration of (M, f) to be interleaved with that of the function-Rips complex Rδ(M•) of any nearby pair (M, f). In particular, our result answers a longstanding question on the related topic of estimating sublevel set persistent homology from finite point samples.
AB - Latschev’s theorem provides sufficient conditions on a metric space M and δ > 0 for the homotopy type of M to agree with that of the Vietoris-Rips complex Rδ(M) of any nearby space M in the Gromov-Hausdorff distance. We prove a persistent version of this theorem, providing sufficient conditions on a pair (M, f : M → RN) and δ > 0 for the persistent homotopy type of the sublevel set filtration of (M, f) to be interleaved with that of the function-Rips complex Rδ(M•) of any nearby pair (M, f). In particular, our result answers a longstanding question on the related topic of estimating sublevel set persistent homology from finite point samples.
KW - Topological data analysis (TDA)
KW - Vietoris-Rips complex
KW - homotopy theory
KW - metric geometry
KW - multi-parameter persistent homology
UR - https://www.scopus.com/pages/publications/105041212098
U2 - 10.4230/LIPIcs.SoCG.2026.82
DO - 10.4230/LIPIcs.SoCG.2026.82
M3 - Conference contribution
AN - SCOPUS:105041212098
T3 - Leibniz International Proceedings in Informatics, LIPIcs
BT - 42nd International Symposium on Computational Geometry, SoCG 2026
A2 - Ahn, Hee-Kap
A2 - Hoffmann, Michael
A2 - Nayyeri, Amir
PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
T2 - 42nd International Symposium on Computational Geometry, SoCG 2026
Y2 - 2 June 2026 through 5 June 2026
ER -