Stability of Persistence Diagrams

By David Cohen-Steiner, Herbert Edelsbrunner, John Harer. Discrete & Computational Geometry, 2005

1,817 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Mathematical Physics, Radiology, Nuclear Medicine and Imaging

View Paper: https://www.semanticscholar.org/paper/aa0ebabd99da5fb367c9e7746ed5ff6f9de03d3a

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Citations in the corpus

Bounds on spectral norms and barcodes (2018, background): “s parametrized by a real parameter forms an algebraic structure called a persistence module which was introduced and studied extensively since the early 2000s in the data analysis community (see e.g. [12, 22, 23, 28, 31, 34, 36, 49, 121]). Recently, persistence modules found applications in symplect”
Morse Theory for Filtrations and Efficient Computation of Persistent Homology (2013, background): “Three fundamental properties account for the importance of persistent homology: (i) being based on algebraic topology, it provides a well understood codification of potentially complicated and/or high dimensional geometric information, (ii) the information it provides is stable with respect to pertu”
Statistical topological data analysis using persistence landscapes (2012, methodology): “For our final stability theorem, we use ideas from Cohen-Steiner et al. (2010). Let f : X → R be a function on a topological space.”
Topological Consistency via Kernel Estimation (2014, methodology): “The validity of these methods is established by using the stability theorem [19] stating that d B (PH ∗ ( f ) , PH ∗ ( ˆ f )) ≤ (cid:107) f − ˆ f ∞ .”
Topology-Preserving Deep Image Segmentation (2019, background): “As a theoretical reassurance, it has been proven that this metric for diagrams is stable, and the loss function Ltopo(f, g) is Lipschitz with regard to the likelihood function f [11].”
Finding the Homology of Submanifolds with High Confidence from Random Samples (2008, methodology): “Finally, it is also worth noting that there is a body of work on persistence homology [7, 20] that seeks alternative topological characterizations of the manifold and its homology.”
Barcodes: The persistent topology of data (2007, background): “Indeed, as shown in [7], barcodes are stable in the presence of noise added to a [Morse] filtration.”
Persistent Homology Analysis of Brain Artery Trees (2014, background): “Many technical results (for example, [12, 13]) all basically say that, under mild conditions, Wp(Dgm0(f),Dgm0(g)) ≤ K||f − g||∞.”
Lipschitz Functions Have Lp-Stable Persistence (2010, methodology): “A breakthrough result is the stability of these diagrams proved in [3].”
An Introduction to Topological Data Analysis: Fundamental and Practical Aspects for Data Scientists (2017, background): “For example we can easily recover the ﬁrst persistence stability result that appeared in the literature (Cohen-Steiner et al., 2005).”
Digital Topology and Geometry in Medical Imaging: A Survey (2015, methodology): “[347] adopted persistent homology [348], [349] for segmentation of papillary muscles and the trabeculae from high resolution cardiac CTs through restoration of topological handles.”
Proximity of persistence modules and their diagrams (2009, methodology): “As mentioned above, although in some speciﬁc scenarios the problem can be easily reduced to some continuous setting, it is not clear that such reductions exist in general.”
Persistent homology and Floer-Novikov theory (2015, background): “sets of elements of (R/Γ)×[0,∞] then we write dB(S,T ) = sup k∈Z dB(Sk,Tk) It is easy to see that in the special case that Γ = {0} the above deﬁnition agrees with the notion of bottleneck distance in [CEH07]. On the Floer complex side, we make the following deﬁnition which is a slight modiﬁcation of”