Proximity of persistence modules and their diagrams

By Frédéric Chazal, David Cohen-Steiner, Marc Glisse, Leonidas J. Guibas, Steve Y. Oudot. SCG '09, 2009

536 citations recorded by the source corpus.

Subfields: Biological Psychiatry, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8fb71b5a8b420ca9cc98d5d6137ffbffa80cbc9a

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Theory and Mathematics

In Mathematical Physics

Citations in the corpus

Decomposition of pointwise finite-dimensional persistence modules (2012, background): “Persistence modules indexed by the natural numbers were introduced in [11] as algebraic objects underlying persistent homology, but other indexing sets are of interest, especially the real numbers—see [1, 2, 3, 4, 8].”
The Structure and Stability of Persistence Modules (2012, background): “We prove the nontrivial result (from [6]) that if two persistence modules are δ-interleaved, then they are connected in the space of persistence modules by a path of length δ.”
Betti numbers in multidimensional persistent homology are stable functions (2013, background): “In these papers, it is gained as a direct consequence of assuming that the filtering functions [20] or the persistence modules [22] are tame.”
Persistent homology and Floer-Novikov theory (2015, methodology): “originally established for the persistence modules associated to “tame” functions on topological spaces in [CEH07]; since then a variety of diﬀerent proofs and generalizations have appeared (see e.g. [CCGGO09], [BL14]), and it now generally understoodas an algebraic statement in the abstract context”
Asymptotic theory for density ridges (2014, background): “Second, as is shown in Genovese et al. (2014), when a topological assumption called tameness is assumed [CohenSteiner, Edelsbrunner and Harer (2007), Chazal et al. (2009, 2012)], then Rh and R have the same topology for small h.”
A roadmap for the computation of persistent homology (2015, result): “This result was generalized in the papers [17,20,26].”
An Introduction to Topological Data Analysis: Fundamental and Practical Aspects for Data Scientists (2017, background): “Under similar but slightly more technical conditions the Reconstruction Theorem can be extended to prove that the sublevel sets are indeed homeomorphic and even isotopic Chazal et al. (2009c, 2008).”
Topological Consistency via Kernel Estimation (2014, background): “This alternating sequence is called ‘interleaving’ and the work in [15] provides means to bound the distance between the persistent homology computed for these two types of ﬁltrations.”
Persistence stability for geometric complexes (2012, background): “Following [4,7] we say that a persistence module V is q-tame if rank(vb a) < ∞ whenever a < b.”
Clique topology reveals intrinsic geometric structure in neural correlations (2015, background): “Further work has broadened this result by developing more complete theoretical tools for the comparison of persistence structures, divorcing their stability from any underlying geometry [24, 25, 26].”
Persistent Convolution: A Topological Framework for AI Alignment Testing and Semantic Space Characterization (2026, unknown): “Stable persistence diagrams can be constructed from q-tame persistence modules (Chazal et al., 2009) (Chazal et al., 2016).”