Computing Persistent Homology

By Afra Zomorodian, Gunnar E. Carlsson. SCG '04, 2004

2,149 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c11e55c089deaefb1fad4b0936921dc94862bab7

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Citations in the corpus

Coverage in sensor networks via persistent homology (2007, methodology): “) The current implementation of Plex computes the dimensions of persistent homology groups (using algorithms as in [8, 21]), which is enough to check whether the homomorphism ι∗ in the criterion of Theorem 3.”
Persistence stability for geometric complexes (2012, background): “One way round this issue is to use topological persistence [11,18], which encodes the homology of the entire nested family Rips(X) = (Rips(X, α))α∈R in a single invariant, the persistence diagram.”
TopologyNet: Topology based deep convolutional and multi-task neural networks for biomolecular property predictions (2017, background): “In this section, we give a brief explanation of persistent homology 45,55 before introducing topological representations of protein-ligand binding interactions and protein stability changes upon mutation.”
Theory of interleavings on categories with a flow (2017, methodology): “Some commonly used topological signatures include persistence diagrams [13], persistence modules [5, 24], Reeb graphs [9, 21], and Mapper [22].”
Morse Theory for Filtrations and Efficient Computation of Persistent Homology (2013, methodology): “The most common algorithm used for computing persistent homology is presented in [37] wherein it is remarked that the worst case complexity is of the same order in time and space as that of computing homology.”
Towards the Augmented Pathologist: Challenges of Explainable-AI in Digital Pathology (2017, background): “A common way to analyze topological structures is to apply persistent homology (Zomorodian and Carlsson, 2005), which identiﬁes cluster, holes and voids.”
Autonomous Hamiltonian flows, Hofer’s geometry and persistence modules (2014, background): “In this section we collect preliminaries on persistence modules, see [5, 11, 12, 14, 16, 25].”
Sample Complexity of Testing the Manifold Hypothesis (2010, background): “There is a significant amount of recent research aimed at understanding topological aspects of data, such its homology [20, 16].”
Persistent homology analysis of protein structure, flexibility and folding (2014, neutral): “Carlsson G. Topology and data.”
Homological scaffolds of brain functional networks (2014, methodology): “The method that we adopt was introduced in references [29,30] and relies on an extension of the metrical persistent homology theory originally introduced by references [31,32].”
Stability of Persistence Diagrams (2005, background): “Recently, Carlsson et al. introduced barcodes, which are persistence diagrams (drawing points in the plane as intervals) for the curvature function of a certain derived space of the shape [3].”
Persistent homology for random fields and complexes (2010, background): “;:::;c j m j g, of H n(K j), for all j0, such that for any 1 km j, i (c j k ) 2f0;c j+1 1 ;c j+1 2 ;:::;c j+1 m j+1 gand i (c j k ) = i (c j k0 ) for k6= k0only if i (c j k ) = 0. 1For details see [39]. imsart-coll ver. 2009/08/13 file: larry.tex date: March 29, 2010 Persistent Homology for Random F”
Proximity of persistence modules and their diagrams (2009, background): “…F A as the one of a family { F α } α ∈ A of vector spaces (or modules over a same commutative ring) indexed by A ⊆ R , together with a family of homomorphisms { f βα : they proved that persistence diagrams can be deﬁned for persistence modules satisfying some tameness condition similar to (3.).”
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, methodology): “There is a ‘snapping principle’ by which our abstractly defined diagrams are seen to agree with the diagrams that are produced by the standard algorithms [17, 25], for example when working with finite simplicial complexes derived from real data.”
Clique topology reveals intrinsic geometric structure in neural correlations (2015, background): “Note that computing low-dimensional ðm≤ 3Þ Betti curves for matrices of size N∼100 is numerically tractable due to recent advances in computational topology (7, 10, 11).”
Topology and data (2009, methodology): “This is the observation made in [64], where the algorithm that we use for computing persistent homology is described in detail.”