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Topological Persistence and Simplification
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TL;DR
This paper formalizes topological persistence in a filtration, using the lifetime of homology classes to distinguish features from noise, and gives algorithms to compute persistence and simplify filtrations. It demonstrates the approach on alpha-complex filtrations of molecular and surface data, including separation of a tunnel in Gramicidin A from shorter-lived cycles.
Key findings
- Persistence is defined for Betti numbers and non-bounding cycles by factoring a cycle group at one filtration stage by the boundary group at a later stage; cycles with short lifetimes are treated as topological noise, while longer-lived ones are features.
- The pairing algorithm matches positive \(k\)-simplices with negative \((k+1)\)-simplices; index-based persistence is \(j-i-1\), and the worst-case running time is \(O(m^3)\), with union-find acceleration for \(k=0\) and \(k=2\) giving amortized \(O(A^{-1}(m))\) per operation.
- On five datasets, total persistence timings ranged from 0.06 s for Gramicidin A to 19.93 s for the Buddha dataset with union-find, versus 30.52 s without it; average collision counts and list lengths were small, and conflicts were rare (basic conflicts 0–1, recursive conflicts 0–212 across datasets).
- For Gramicidin A, removing 1-cycles with persistence below 2688 isolated the molecular tunnel from other topological attributes.
Methods
- The work studies finite simplicial complexes in \(\mathbb{R}^3\), especially alpha complexes of Delaunay triangulations, ordered by an age filter induced by growing weighted balls.
- Persistence is formulated algebraically as \(H^{\ell,p}_k = Z^\ell_k / (B^{\ell+p}_k \cap Z^\ell_k)\), and an abstract pairing algorithm uses a hash table and cycle search to pair simplex births with deaths; simplification is performed by reordering simplices via migration, with conflict resolution or diminution.
- Experiments implemented the algorithms with alpha-shapes software on five datasets: three weighted molecular structures and two unweighted macroscopic surfaces; timings were collected on a 266 MHz Pentium II with 128 MB RAM under Solaris 8.
Limitations
- Stated: the detailed analysis of the 1-cycle computation is left open, and the authors ask for a different, more efficient algorithm; worst-case cubic behavior contrasts with observed near-linear timing, and conflicts, though rare, require handling.
- Stated: conflict resolution by subdivision may not correspond to the alpha-complex growth model, while conflict diminution changes when cycles affect Betti numbers.
- Evident from the text: the experimental evidence covers only five datasets and a prototype implementation; time-based persistence is not developed further, and the setting is restricted to \(\mathbb{R}^3\) complexes with \(\mathbb{Z}_2\) coefficients.
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“The primary topological feature of this data is a tunnel that runs through the molecule, as shown in Fig. 17.”
§ Discrete & Computational Geometry > Feature Detection.
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“The image of the homomorphism is isomorphic to the p-persistent homology group of K ℓ , im η ℓ, p k H ℓ, p k .”
§ Discrete & Computational Geometry > Algebraic Formulation.
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“Then Z ℓ k ∩ B ℓ+ p-1 k = Z ℓ k ∩ B ℓ+ p k , and hence β ℓ, p k = β ℓ, p-1 k”
§ Discrete & Computational Geometry > Abstract Algorithm.
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“After accelerating with union-find, the slowest portion of the algorithm is adding 1-cycles.”
§ Discrete & Computational Geometry > Timings.
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“In other words, increasing p by one shortens the persistence of all nonbounding cycles by one.”
§ Discrete & Computational Geometry > Algebraic Formulation.
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“k-Triangle Lemma.The number of k-triangles containing (ℓ, p) in the index-persistence plane is β ℓ, p k .”
§ Discrete & Computational Geometry > Abstract Algorithm.
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“While there is considerable topological noise at p = 0, a simplification process which eliminates 1cycles of persistence less than 2688 succeeds in separating the tunnel from the remaining topological attributes detected by measuring homology.”
§ Discrete & Computational Geometry > Feature Detection.
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“We will see in Section 3 that there exists a pairing between positive k-simplices and negative (k + 1)-simplices.”
§ Discrete & Computational Geometry > Incremental Algorithm.
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“If follows that β ℓ, p k = β ℓ, p-1 k -1.”
§ Discrete & Computational Geometry > Abstract Algorithm.
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“Monotonicity Lemma.β ℓ, p k ≤ β ℓ , p k whenever p ≤ p and ℓ ≤ ℓ ≤ ℓ + ( pp ).”
§ Discrete & Computational Geometry > Abstract Algorithm.
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“The persistence of that k-cycle is one short of the difference between indices, ji -1.”
§ Discrete & Computational Geometry > Abstract Algorithm.
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“We can write each tunnel as a sum of tunnels in a basis, and β 1 is the size of the basis.”
§ Discrete & Computational Geometry > Homology Groups.
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“As before, β 2 is the size of a basis of voids, which is equal to the number of voids.”
§ Discrete & Computational Geometry > Homology Groups.
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“All Betti numbers are non-negative so pos k ≥ neg k+1 for all ℓ.”
§ Discrete & Computational Geometry > Incremental Algorithm.
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“All we guarantee is that d is homologous to the sum of cycles represented by the simplices in the list, and that the list contains the youngest simplex in (d), which is σ i as above.”
§ Discrete & Computational Geometry > Data Structure.
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“Basic and recursive conflicts exist in practice, but are rather rare, as shown in Section 6.”
§ Discrete & Computational Geometry > Conflict Lemma. All conflicts have type (1, 2).
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“For each simplex σ ∈ D, there is a unique birth time α(σ ) defined such that σ ∈ K (α) iff α ≥ α(σ ).”
§ Discrete & Computational Geometry > Age Filters.
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“The pairs (su, suv) and (tw, stu) in Fig. 6 constitute a conflict of type (1,2) and show that conflicts do occur, although our experiments in Section 6 suggest that they are rather rare.”
§ Discrete & Computational Geometry > Conflicts.
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“The results suggest a possible linear dependence of the running time on the size of the data, which is substantially faster than the cubic dependence proved in Section 4.”
§ Discrete & Computational Geometry > Timings.
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“Tables 3 and 4 provide evidence that this might indeed be the case.”
§ Discrete & Computational Geometry > Statistics.
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“Furthermore, if σ ∈ K , then all faces of σ are simplices in K .”
§ Discrete & Computational Geometry > Figure 1 illustrates a two-dimensional example of this construction.
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- The authors state this as an assumption: The point set is in general position, so at most four (three in R^2) Voronoi regions can have a non-empty common intersection.
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“The incremental algorithm in Section 2 asserts that β ℓ k is the number of k-intervals that contain index ℓ on the axis.”
§ Discrete & Computational Geometry > Abstract Algorithm.
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“We do not show statistics for 0-cycles in Table 4 as every 0-cycle is represented by a list of length 2.”
§ Discrete & Computational Geometry > Statistics.
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“Any two simplices in K are either disjoint or they intersect in a common face, which is a simplex of smaller dimension.”
§ Discrete & Computational Geometry > Figure 1 illustrates a two-dimensional example of this construction.
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“As substantiated in Table 2, this improvement subsumes adding 0-and 2-cycles in the marking process, shrinking the time for these to steps to essentially nothing.”
§ Discrete & Computational Geometry > Timings.
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“Table 5 shows that the average persistence is considerably larger than the average number of collisions and list length.”
§ Discrete & Computational Geometry > Statistics.
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“The total algorithm runs in time at most O( p 2 i ), which is at most O(m 3 ).”
§ Discrete & Computational Geometry > Collision Lemma.
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“Our motivation for formulating persistent homology in (3) was to eliminate cycles with low persistence.”
§ Discrete & Computational Geometry > Lazy Migration.
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“There are 4 2 = 6 possible types of conflicts, each identified by the pair (dim σ i , dim σ h ) of the dimensions of the main participants.”
§ Discrete & Computational Geometry > Conflicts.
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“The sequence of iso-surfaces obtained by increasing α represents a growth process similar to that represented by a filtration.”
§ Discrete & Computational Geometry > Approach and Results.
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“As cx is not a face of abx, we have removed the conflict and preserved the Betti numbers of a refined filtration.”
§ Discrete & Computational Geometry > Conflict Resolution.
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“By complementarity and duality, there are no conflicts of types (1, 3) and (2, 3).”
§ Discrete & Computational Geometry > Conflict Lemma. All conflicts have type (1, 2).
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“We can explain the apparently linear running time documented in Table 2 by showing that the average number of collisions and the average list length are both constant.”
§ Discrete & Computational Geometry > Statistics.
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“The cycle search continues until it finds a collision-free cycle homologous to d, and the Collision Lemma implies that that cycle has the correct youngest positive simplex.”
§ Discrete & Computational Geometry > Collision Lemma.
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“A single collision takes time at most O( p i ), and the entire search for σ i takes time at most O( p 2 i ).”
§ Discrete & Computational Geometry > Collision Lemma.
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“Finally, we note that Robins has independently formulated an alternate but equivalent definition of persistence [10].”
§ Discrete & Computational Geometry > Algebraic Formulation.
What the sources say
- Not matchedcites [10] 10. V. Robins. Computing topology at multiple resolutions: foundations and applications to fractals and dynamics. Ph.D. thesis, Department of Applied Mathematics, University of Colorado, June 2000.
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“The ordering of the new simplices specified by subdivision in Fig. 13 might not have a corresponding set of weighted balls that would generate the filtration under the growth model.”
§ Discrete & Computational Geometry > Conflict Resolution.
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- Rests on an assumption: In alpha complex filtrations, simplices are ordered according to a particular growth model.
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In the paper
“Recall that the number of collisions and the length of lists is bounded from above by the persistence of cycles.”
§ Discrete & Computational Geometry > Statistics.
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“Specifically, simplices in a filter correspond to critical points of a density function.”
§ Discrete & Computational Geometry > Approach and Results.
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In the paper
“Hence, i contains fewer than (k + 2) p i k-simplices, and similarly g contains fewer than (k + 2) p g < (k + 2) p i k-simplices.”
§ Discrete & Computational Geometry > Collision Lemma.
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“For k = 0, each set is the vertex set of a connected component.Each set has exactly one yet unpaired vertex, namely the oldest one in the component.”
§ Discrete & Computational Geometry > Collision Lemma.
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“As a consequence of the formulation, the lifetime of every cycle is reduced regardless of its persistence, leading to the creation of k-triangles.”
§ Discrete & Computational Geometry > Lazy Migration.
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“The problem of topological simplification was also approached by El-Sana and Varshney [5] using alpha shape inspired ideas of geometric growth.”
§ Discrete & Computational Geometry > Approach and Results.
What the sources say
- Not checked — no full text of the sourcecites [5] 5. J. El-Sana and A. Varshney. Topology simplification for polygonal virtual environments. IEEE Trans. Visual. Comput. Graphics, 4:133-144, 1998. source
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In the paper
“We cannot achieve the same acceleration for 1-cycles using this method, however, as there can be multiple unpaired positive edges at any time.”
§ Discrete & Computational Geometry > Collision Lemma.
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In the paper
“The number of collisions is at most the number of positive k-simplices strictly between σ i and σ j , which is less than p i .”
§ Discrete & Computational Geometry > Collision Lemma.
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In the paper
“We interpret these intervals as k-intervals of a simplified version of the original filtration.”
§ Discrete & Computational Geometry > 5. Simplification
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In the paper
“We may resolve a conflict by subdivision.Here, we achieve the correct Betti numbers for a refined complex and its corresponding filtrations.”
§ Discrete & Computational Geometry > Conflict Resolution.
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In the paper
“At the time we move it, the face is already collocated with its matching negative simplex, and the two cancel each other's contributions.”
§ Discrete & Computational Geometry > 5. Simplification
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In the paper
“A collision happens at T[g] only if σ g already forms a pair, which implies its k-interval [g, h) is contained inside [i, j).”
§ Discrete & Computational Geometry > Collision Lemma.
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The authors state this as an assumption.
The point set is in general position, so at most four (three in R^2) Voronoi regions can have a non-empty common intersection.
“We assume general position, where at most four (three in R 2 ) Voronoi regions can have a non-empty common intersection.”
The paper relies on this without support.
Complexes embedded in three-dimensional Euclidean space have no torsion.
“There is no torsion in complexes embedded in three-dimensional Euclidean space, so the homology groups are vector spaces.”
The paper relies on this without support.
In alpha complex filtrations, simplices are ordered according to a particular growth model.
“For example, in alpha complex filtrations, simplices are ordered according to a particular growth model.”
The paper relies on this without support.
The five data sets used in the experiments are representative of their respective classes of molecular structures and macroscopic shapes.
“We have applied the software to a variety of datasets and show the results for five representative sets in this section.”
The paper relies on this without support.
The Gramicidin A dataset contains the time-averaged molecular dynamics structure of Gramicidin A, a peptide that forms a channel for ion and water movement across lipid membranes.
“This dataset contains the time-averaged molecular dynamics structure of Gramicidin A, a peptide that forms a channel for ion and water movement across lipid membranes.”
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In the citing paper
“In other words, the authors prove constructively that persistent homology groups of subcomplexes of S 3 , if computed over Z 2 coefficients, have a simple description in terms of a set of intervals.”
From Computing Persistent Homology (2005) · cites this paper as [11]
Experimental — accuracy not yet measuredThe source backs part of the claim, not all of it — or backs it for a different population, condition or comparison.
In the citing paper
“Surprisingly, the authors show that these intervals allowed the correct computation of the rank of persistent homology groups.”
From Computing Persistent Homology (2005) · cites this paper as [11]
Experimental — accuracy not yet measuredThe source backs part of the claim, not all of it — or backs it for a different population, condition or comparison.
In the citing paper
“The authors also provide an algorithm that worked only for spaces that were subcomplexes of S 3 over Z 2 coefficients.”
From Computing Persistent Homology (2005) · cites this paper as [11]
Experimental — accuracy not yet measuredThe cited source says what the claim attributes to it.
In the citing paper
“Therefore, the P -intervals discussed here are precisely the so-called k -intervals utilized in [11] to describe persistent Z 2-homology.”
From Computing Persistent Homology (2005) · cites this paper as [11]
The cited source says what the claim attributes to it.
In the citing paper
“Lemma 3.1. Let T be the set of triangles defined by P -intervals for the k-dimensional persistence module . The rank β l , p k of H l , p k is the number of triangles in T containing the point ( l , p ) .”
From Computing Persistent Homology (2005) · cites this paper as [11]
The cited source says what the claim attributes to it.
In the citing paper
“This algorithm is a generalized version of the pairing algorithm shown in [11].”
From Computing Persistent Homology (2005) · cites this paper as [11]
The cited source says what the claim attributes to it.
In the citing paper
“Then im η i , p k ≃ H i , p k [11], [18].”
From Computing Persistent Homology (2005) · cites this paper as [11]
In the citing paper
“As a result, persistent homology can capture topological structures continuously over a range of spatial scales.”
From Quantitative Toxicity Prediction Using Topology Based Multitask Deep Neural Networks (2018) · cites this paper as [17]
In the citing paper
“Unlike commonly used computational homology which results in truly metric free representations, persistent homology embeds geometric information in topological invariants, e.g., Betti numbers, so that 'birth' and 'death' of isolated components, rings, and cavities can be monitored at all geometric scales by topological measurements.”
From Quantitative Toxicity Prediction Using Topology Based Multitask Deep Neural Networks (2018) · cites this paper as [17]
In the citing paper
“Persistent homology22–25—a new branch of algebraic topology—is able to bridge geometry and topology, leading to a new efficient approach for the simplification of biological structural complexity26–31; however, it neglects critical chemical/biological information when it is directly applied to complex biomolecular structures.”
From A topology-based network tree for the prediction of protein–protein binding affinity changes following mutation (2020) · cites this paper as [23]
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
Each contradiction between two passages listed here was confirmed by an independent check.
Induced homology maps are described as injective despite cycle deaths
MinorAn injective map from an earlier homology group cannot send one of its nonzero classes to zero. The paper also says that later negative simplices cancel classes created earlier, so the homology maps it describes cannot generally be injective.
“The p-persistent homology groups can also be defined using injective homomorphisms between ordinary homology groups.”
“Note that as we increase p, negative simplices cancel positive simplices earlier in the filtration.”
One reported average list length exceeds average persistence
MinorFor G's 2-cycles, Table 4 gives an average list length of 4.86, while Table 5 gives an average persistence of 4.47. Thus the stated comparison does not hold for that dataset and cycle dimension.
“Table 5 shows that the average persistence is considerably larger than the average number of collisions and list length.”
“| G | 10 | 2.47 | 2.22 | 111 | 4.86 | 2.08 |”
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Metadata
- Authors
- Edelsbrunner, Letscher, & Zomorodian
- Affiliations
- Duke University (US)
- Oklahoma State University (US)
- University of Illinois Urbana-Champaign (US)
- Venue
- Discrete & Computational Geometry
- Published
- 2002-11
- Volume
- 28
- Issue
- 4
- Pages
- 511-533
- ISSN
- 0179-5376 (ISSN-L)
- 1432-0444
- OpenAlex
- W3013843370
- Cited by
- 1,779
- References
- 12
- Citations by year
- 2026: 130 · 2025: 152 · 2024: 125 · 2023: 94 · 2022: 91 · 2021: 139 · 2020: 106 · 2019: 131 · 2018: 101 · 2017: 72 · 2016: 89 · 2015: 70 · 2014: 74 · 2013: 47 · 2012: 63
- FWCI
- 13.65
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Licence
- www.springer.com/tdm (tdm)
- Locations
- Discrete & Computational Geometry (publishedVersion)
- Funders
- National Science Foundation (9712088)
- Army Research Office
- Topics
- Physical Sciences › Computer Science › Computational Theory and Mathematics › Topological and Geometric Data Analysis
- Physical Sciences › Computer Science › Computer Vision and Pattern Recognition › Digital Image Processing Techniques
- Physical Sciences › Mathematics › Geometry and Topology › Geometric and Algebraic Topology
- Keywords
- topological persistence, topological simplification, simplicial complexes, filtration, computational topology
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