16 Search Results for "Lesnick, Michael"

Document
DOI: 10.4230/LIPIcs.SoCG.2026.5
Bifunction and Interlevel Delaunay Trifiltrations

Authors: Ángel Javier Alonso, Michael Kerber, Tung Lam, Michael Lesnick, and Abhishek Rathod

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract
A key property of the Delaunay filtration is that it is topologically (i.e., weakly) equivalent to the offset (union-of-balls) filtration. Recently, this filtration has been extended to point clouds equipped with an ℝ-valued function, yielding a computable 2-parameter filtration that satisfies an analogous weak equivalence. Motivated in part by the study of time-varying data, we introduce a 3-parameter extension of the Delaunay filtration for point clouds equipped with an ℝ²-valued function, also satisfying an analogous weak equivalence. For a point cloud X ⊂ ℝ^d, our trifiltration has size O(|X|^{⌈(d+1)/2⌉+1}). We present an algorithm that computes this trifiltration in time O(|X|^{⌈d/2⌉+2}), together with an implementation. Our experiments demonstrate that the implementation can handle thousands of points in ℝ³, with memory growth that is nearly linear.
Cite as

Ángel Javier Alonso, Michael Kerber, Tung Lam, Michael Lesnick, and Abhishek Rathod. Bifunction and Interlevel Delaunay Trifiltrations. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 5:1-5:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{alonso_et_al:LIPIcs.SoCG.2026.5,
  author =	{Alonso, \'{A}ngel Javier and Kerber, Michael and Lam, Tung and Lesnick, Michael and Rathod, Abhishek},
  title =	{{Bifunction and Interlevel Delaunay Trifiltrations}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{5:1--5:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.5},
  URN =		{urn:nbn:de:0030-drops-258118},
  doi =		{10.4230/LIPIcs.SoCG.2026.5},
  annote =	{Keywords: Delaunay triangulation, Multiparameter persistent homology, Interlevel, Bowyer-Watson}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2025.29
Sparsification of the Generalized Persistence Diagrams for Scalability Through Gradient Descent

Authors: Mathieu Carrière, Seunghyun Kim, and Woojin Kim

Published in: LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025)

Abstract
The generalized persistence diagram (GPD) is a natural extension of the classical persistence barcode to the setting of multi-parameter persistence and beyond. The GPD is defined as an integer-valued function whose domain is the set of intervals in the indexing poset of a persistence module, and is known to be able to capture richer topological information than its single-parameter counterpart. However, computing the GPD is computationally prohibitive due to the sheer size of the interval set. Restricting the GPD to a subset of intervals provides a way to manage this complexity, compromising discriminating power to some extent. However, identifying and computing an effective restriction of the domain that minimizes the loss of discriminating power remains an open challenge. In this work, we introduce a novel method for optimizing the domain of the GPD through gradient descent optimization. To achieve this, we introduce a loss function tailored to optimize the selection of intervals, balancing computational efficiency and discriminative accuracy. The design of the loss function is based on the known erosion stability property of the GPD. We showcase the efficiency of our sparsification method for dataset classification in supervised machine learning. Experimental results demonstrate that our sparsification method significantly reduces the time required for computing the GPDs associated to several datasets, while maintaining classification accuracies comparable to those achieved using full GPDs. Our method thus opens the way for the use of GPD-based methods to applications at an unprecedented scale.
Cite as

Mathieu Carrière, Seunghyun Kim, and Woojin Kim. Sparsification of the Generalized Persistence Diagrams for Scalability Through Gradient Descent. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 29:1-29:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{carriere_et_al:LIPIcs.SoCG.2025.29,
  author =	{Carri\`{e}re, Mathieu and Kim, Seunghyun and Kim, Woojin},
  title =	{{Sparsification of the Generalized Persistence Diagrams for Scalability Through Gradient Descent}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{29:1--29:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-370-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{332},
  editor =	{Aichholzer, Oswin and Wang, Haitao},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.29},
  URN =		{urn:nbn:de:0030-drops-231810},
  doi =		{10.4230/LIPIcs.SoCG.2025.29},
  annote =	{Keywords: Multi-parameter persistent homology, Generalized persistence diagram, Generalized rank invariant, Non-convex optimization, Gradient descent}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2025.58
Tracking the Persistence of Harmonic Chains: Barcode and Stability

Authors: Tao Hou, Salman Parsa, and Bei Wang

Published in: LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025)

Abstract
The persistence barcode is a topological descriptor of data that plays a fundamental role in topological data analysis. Given a filtration of data, the persistence barcode tracks the evolution of its homology groups. In this paper, we introduce a new type of barcode, called the harmonic chain barcode, which tracks the evolution of harmonic chains. In addition, we show that the harmonic chain barcode is stable. Given a filtration of a simplicial complex of size m, we present an algorithm to compute its harmonic chain barcode in O(m³) time. Consequently, the harmonic chain barcode can enrich the family of topological descriptors in applications where a persistence barcode is applicable, such as feature vectorization and machine learning.
Cite as

Tao Hou, Salman Parsa, and Bei Wang. Tracking the Persistence of Harmonic Chains: Barcode and Stability. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 58:1-58:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{hou_et_al:LIPIcs.SoCG.2025.58,
  author =	{Hou, Tao and Parsa, Salman and Wang, Bei},
  title =	{{Tracking the Persistence of Harmonic Chains: Barcode and Stability}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{58:1--58:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-370-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{332},
  editor =	{Aichholzer, Oswin and Wang, Haitao},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.58},
  URN =		{urn:nbn:de:0030-drops-232100},
  doi =		{10.4230/LIPIcs.SoCG.2025.58},
  annote =	{Keywords: Persistent homology, harmonic chains, topological data analysis}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2025.6
A Sparse Multicover Bifiltration of Linear Size

Authors: Ángel Javier Alonso

Published in: LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025)

Abstract
The k-cover of a point cloud X of ℝ^d at radius r is the set of all those points within distance r of at least k points of X. By varying r and k we obtain a two-parameter filtration known as the multicover bifiltration. This bifiltration has received attention recently due to being choice-free and robust to outliers. However, it is hard to compute: the smallest known equivalent simplicial bifiltration has O(|X|^{d+1}) simplices. In this paper we introduce a (1+ε)-approximation of the multicover bifiltration of linear size O(|X|), for fixed d and ε. The methods also apply to the subdivision Rips bifiltration on metric spaces of bounded doubling dimension yielding analogous results.
Cite as

Ángel Javier Alonso. A Sparse Multicover Bifiltration of Linear Size. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 6:1-6:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{alonso:LIPIcs.SoCG.2025.6,
  author =	{Alonso, \'{A}ngel Javier},
  title =	{{A Sparse Multicover Bifiltration of Linear Size}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{6:1--6:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-370-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{332},
  editor =	{Aichholzer, Oswin and Wang, Haitao},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.6},
  URN =		{urn:nbn:de:0030-drops-231587},
  doi =		{10.4230/LIPIcs.SoCG.2025.6},
  annote =	{Keywords: Multicover, Approximation, Sparsification, Multiparameter persistence}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2025.14
Extremal Betti Numbers and Persistence in Flag Complexes

Authors: Lies Beers and Magnus Bakke Botnan

Published in: LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025)

Abstract
We investigate several problems concerning extremal Betti numbers and persistence in filtrations of flag complexes. For graphs on n vertices, we show that β_k(X(G)) is maximal when G = 𝒯_{n,k+1}, the Turán graph on k+1 partition classes, where X(G) denotes the flag complex of G. Building on this, we construct an edgewise (one edge at a time) filtration 𝒢 = G₁ ⊆ ⋯ ⊆ 𝒯_{n,k+1} for which β_k(X(G_i)) is maximal for all graphs on n vertices and i edges. Moreover, the persistence barcode ℬ_k(X(G)) achieves a maximal number of intervals, and total persistence, among all edgewise filtrations with |E(𝒯_{n,k+1})| edges. For k = 1, we consider edgewise filtrations of the complete graph K_n. We show that the maximal number of intervals in the persistence barcode is obtained precisely when G_{⌈n/2⌉ ⋅ ⌊n/2⌋} = 𝒯_{n,2}. Among such filtrations, we characterize those achieving maximal total persistence. We further show that no filtration can optimize β₁(X(G_i)) for all i, and conjecture that our filtrations maximize the total persistence over all edgewise filtrations of K_n.
Cite as

Lies Beers and Magnus Bakke Botnan. Extremal Betti Numbers and Persistence in Flag Complexes. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 14:1-14:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{beers_et_al:LIPIcs.SoCG.2025.14,
  author =	{Beers, Lies and Bakke Botnan, Magnus},
  title =	{{Extremal Betti Numbers and Persistence in Flag Complexes}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{14:1--14:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-370-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{332},
  editor =	{Aichholzer, Oswin and Wang, Haitao},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.14},
  URN =		{urn:nbn:de:0030-drops-231668},
  doi =		{10.4230/LIPIcs.SoCG.2025.14},
  annote =	{Keywords: Topological data analysis, Extremal graph theory}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2025.41
Decomposing Multiparameter Persistence Modules

Authors: Tamal K. Dey, Jan Jendrysiak, and Michael Kerber

Published in: LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025)

Abstract
Dey and Xin (J.Appl.Comput.Top., 2022) describe an algorithm to decompose finitely presented multiparameter persistence modules using a matrix reduction algorithm. Their algorithm only works for modules whose generators and relations are distinctly graded. We extend their approach to work on all finitely presented modules and introduce several improvements that lead to significant speed-ups in practice. Our algorithm is fixed-parameter tractable with respect to the maximal number of relations of the same degree and with further optimisation we obtain an O(n³) time algorithm for interval-decomposable modules. In particular, we can decide interval-decomposability in this time. As a by-product to the proofs of correctness we develop a theory of parameter restriction for persistence modules. Our algorithm is implemented as a software library aida, the first to enable the decomposition of large inputs. We show its capabilities via extensive experimental evaluation.
Cite as

Tamal K. Dey, Jan Jendrysiak, and Michael Kerber. Decomposing Multiparameter Persistence Modules. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 41:1-41:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{dey_et_al:LIPIcs.SoCG.2025.41,
  author =	{Dey, Tamal K. and Jendrysiak, Jan and Kerber, Michael},
  title =	{{Decomposing Multiparameter Persistence Modules}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{41:1--41:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-370-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{332},
  editor =	{Aichholzer, Oswin and Wang, Haitao},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.41},
  URN =		{urn:nbn:de:0030-drops-231939},
  doi =		{10.4230/LIPIcs.SoCG.2025.41},
  annote =	{Keywords: Topological Data Analysis, Multiparameter Persistence Modules, Persistence, Decomposition}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2025.35
A Theory of Sub-Barcodes

Authors: Oliver A. Chubet, Kirk P. Gardner, and Donald R. Sheehy

Published in: LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025)

Abstract
The primary tool in topological data analysis (TDA) is persistent homology, which involves computing a barcode - often from point-cloud or scalar field data - that serves as a topological signature for the underlying function. In this work, we introduce sub-barcodes and show how they arise naturally from factorizations of persistence module homomorphisms. We show that, as a partial order induced by factorizations, the relation of being a sub-barcode is strictly stronger than the rank invariant, and we apply sub-barcode theory to the problem of inferring information about the barcode of an unknown Lipschitz function from samples. The advantage of this approach is that it permits strong guarantees - with no noise - while requiring no sampling assumptions, and the resulting barcode is guaranteed to be a sub-barcode of every Lipschitz function that agrees with the data. We also present an algorithmic theory that allows for the efficient approximation of sub-barcodes using filtered Delaunay triangulations for Euclidean inputs.
Cite as

Oliver A. Chubet, Kirk P. Gardner, and Donald R. Sheehy. A Theory of Sub-Barcodes. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 35:1-35:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{chubet_et_al:LIPIcs.SoCG.2025.35,
  author =	{Chubet, Oliver A. and Gardner, Kirk P. and Sheehy, Donald R.},
  title =	{{A Theory of Sub-Barcodes}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{35:1--35:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-370-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{332},
  editor =	{Aichholzer, Oswin and Wang, Haitao},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.35},
  URN =		{urn:nbn:de:0030-drops-231873},
  doi =		{10.4230/LIPIcs.SoCG.2025.35},
  annote =	{Keywords: Topology, Topological Data Analysis, Persistent Homology, Persistence Modules, Barcodes, Sub-barcodes, Factorizations, Lipschitz Extensions}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2025.64
Super-Polynomial Growth of the Generalized Persistence Diagram

Authors: Donghan Kim, Woojin Kim, and Wonjun Lee

Published in: LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025)

Abstract
The Generalized Persistence Diagram (GPD) for multi-parameter persistence naturally extends the classical notion of persistence diagram for one-parameter persistence. However, unlike its classical counterpart, computing the GPD remains a significant challenge. The main hurdle is that, while the GPD is defined as the Möbius inversion of the Generalized Rank Invariant (GRI), computing the GRI is intractable due to the formidable size of its domain, i.e., the set of all connected and convex subsets in a finite grid in ℝ^d with d ≥ 2. This computational intractability suggests seeking alternative approaches to computing the GPD. In order to study the complexity associated to computing the GPD, it is useful to consider its classical one-parameter counterpart, where for a filtration of a simplicial complex with n simplices, its persistence diagram contains at most n points. This observation leads to the question: Given a d-parameter simplicial filtration, could the cardinality of its GPD (specifically, the support of the GPD) also be bounded by a polynomial in the number of simplices in the filtration? This is the case for d = 1, where we compute the persistence diagram directly at the simplicial filtration level. If this were also the case for d ≥ 2, it might be possible to compute the GPD directly and much more efficiently without relying on the GRI. We show that the answer to the question above is negative, demonstrating the inherent difficulty of computing the GPD. More specifically, we construct a sequence of d-parameter simplicial filtrations where the cardinalities of their GPDs are not bounded by any polynomial in the number of simplices. Furthermore, we show that several commonly used methods for constructing multi-parameter filtrations can give rise to such "wild" filtrations.
Cite as

Donghan Kim, Woojin Kim, and Wonjun Lee. Super-Polynomial Growth of the Generalized Persistence Diagram. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 64:1-64:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{kim_et_al:LIPIcs.SoCG.2025.64,
  author =	{Kim, Donghan and Kim, Woojin and Lee, Wonjun},
  title =	{{Super-Polynomial Growth of the Generalized Persistence Diagram}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{64:1--64:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-370-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{332},
  editor =	{Aichholzer, Oswin and Wang, Haitao},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.64},
  URN =		{urn:nbn:de:0030-drops-232162},
  doi =		{10.4230/LIPIcs.SoCG.2025.64},
  annote =	{Keywords: Persistent homology, M\"{o}bius inversion, Multiparameter persistence, Generalized persistence diagram, Generalized rank invariant}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2025.69
Computing Betti Tables and Minimal Presentations of Zero-Dimensional Persistent Homology

Authors: Dmitriy Morozov and Luis Scoccola

Published in: LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025)

Abstract
The Betti tables of a multigraded module encode the grades at which there is an algebraic change in the module. Multigraded modules show up in many areas of pure and applied mathematics, and in particular in topological data analysis, where they are known as persistence modules, and where their Betti tables describe the places at which the homology of filtered simplicial complexes changes. Although Betti tables of singly and bigraded modules are already being used in applications of topological data analysis, their computation in the bigraded case (which relies on an algorithm that is cubic in the size of the filtered simplicial complex) is a bottleneck when working with large datasets. We show that, in the special case of 0-dimensional homology (relevant for clustering and graph classification) Betti tables of bigraded modules can be computed in log-linear time. We also consider the problem of computing minimal presentations, and show that minimal presentations of 0-dimensional persistent homology can be computed in quadratic time, regardless of the grading poset.
Cite as

Dmitriy Morozov and Luis Scoccola. Computing Betti Tables and Minimal Presentations of Zero-Dimensional Persistent Homology. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 69:1-69:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{morozov_et_al:LIPIcs.SoCG.2025.69,
  author =	{Morozov, Dmitriy and Scoccola, Luis},
  title =	{{Computing Betti Tables and Minimal Presentations of Zero-Dimensional Persistent Homology}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{69:1--69:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-370-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{332},
  editor =	{Aichholzer, Oswin and Wang, Haitao},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.69},
  URN =		{urn:nbn:de:0030-drops-232219},
  doi =		{10.4230/LIPIcs.SoCG.2025.69},
  annote =	{Keywords: Multiparameter persistence, Zero-dimensional homology, Minimal presentation, Betti table}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2023.15
Efficient Two-Parameter Persistence Computation via Cohomology

Authors: Ulrich Bauer, Fabian Lenzen, and Michael Lesnick

Published in: LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023)

Abstract
Clearing is a simple but effective optimization for the standard algorithm of persistent homology (ph), which dramatically improves the speed and scalability of ph computations for Vietoris-Rips filtrations. Due to the quick growth of the boundary matrices of a Vietoris-Rips filtration with increasing dimension, clearing is only effective when used in conjunction with a dual (cohomological) variant of the standard algorithm. This approach has not previously been applied successfully to the computation of two-parameter ph. We introduce a cohomological algorithm for computing minimal free resolutions of two-parameter ph that allows for clearing. To derive our algorithm, we extend the duality principles which underlie the one-parameter approach to the two-parameter setting. We provide an implementation and report experimental run times for function-Rips filtrations. Our method is faster than the current state-of-the-art by a factor of up to 20.
Cite as

Ulrich Bauer, Fabian Lenzen, and Michael Lesnick. Efficient Two-Parameter Persistence Computation via Cohomology. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 15:1-15:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bauer_et_al:LIPIcs.SoCG.2023.15,
  author =	{Bauer, Ulrich and Lenzen, Fabian and Lesnick, Michael},
  title =	{{Efficient Two-Parameter Persistence Computation via Cohomology}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{15:1--15:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-273-0},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{258},
  editor =	{Chambers, Erin W. and Gudmundsson, Joachim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.15},
  URN =		{urn:nbn:de:0030-drops-178656},
  doi =		{10.4230/LIPIcs.SoCG.2023.15},
  annote =	{Keywords: Persistent homology, persistent cohomology, two-parameter persistence, clearing}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2022.24
The Universal 𝓁^p-Metric on Merge Trees

Authors: Robert Cardona, Justin Curry, Tung Lam, and Michael Lesnick

Published in: LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022)

Abstract
Adapting a definition given by Bjerkevik and Lesnick for multiparameter persistence modules, we introduce an 𝓁^p-type extension of the interleaving distance on merge trees. We show that our distance is a metric, and that it upper-bounds the p-Wasserstein distance between the associated barcodes. For each p ∈ [1,∞], we prove that this distance is stable with respect to cellular sublevel filtrations and that it is the universal (i.e., largest) distance satisfying this stability property. In the p = ∞ case, this gives a novel proof of universality for the interleaving distance on merge trees.
Cite as

Robert Cardona, Justin Curry, Tung Lam, and Michael Lesnick. The Universal 𝓁^p-Metric on Merge Trees. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 24:1-24:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{cardona_et_al:LIPIcs.SoCG.2022.24,
  author =	{Cardona, Robert and Curry, Justin and Lam, Tung and Lesnick, Michael},
  title =	{{The Universal 𝓁^p-Metric on Merge Trees}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{24:1--24:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-227-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{224},
  editor =	{Goaoc, Xavier and Kerber, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.24},
  URN =		{urn:nbn:de:0030-drops-160325},
  doi =		{10.4230/LIPIcs.SoCG.2022.24},
  annote =	{Keywords: merge trees, hierarchical clustering, persistent homology, Wasserstein distances, interleavings}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2022.58
The Degree-Rips Complexes of an Annulus with Outliers

Authors: Alexander Rolle

Published in: LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022)

Abstract
The degree-Rips bifiltration is the most computable of the parameter-free, density-sensitive bifiltrations in topological data analysis. It is known that this construction is stable to small perturbations of the input data, but its robustness to outliers is not well understood. In recent work, Blumberg-Lesnick prove a result in this direction using the Prokhorov distance and homotopy interleavings. Based on experimental evaluation, they argue that a more refined approach is desirable, and suggest the framework of homology inference. Motivated by these experiments, we consider a probability measure that is uniform with high density on an annulus, and uniform with low density on the disc inside the annulus. We compute the degree-Rips complexes of this probability space up to homotopy type, using the Adamaszek-Adams computation of the Vietoris-Rips complexes of the circle. These degree-Rips complexes are the limit objects for the Blumberg-Lesnick experiments. We argue that the homology inference approach has strong explanatory power in this case, and suggest studying the limit objects directly as a strategy for further work.
Cite as

Alexander Rolle. The Degree-Rips Complexes of an Annulus with Outliers. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 58:1-58:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{rolle:LIPIcs.SoCG.2022.58,
  author =	{Rolle, Alexander},
  title =	{{The Degree-Rips Complexes of an Annulus with Outliers}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{58:1--58:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-227-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{224},
  editor =	{Goaoc, Xavier and Kerber, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.58},
  URN =		{urn:nbn:de:0030-drops-160664},
  doi =		{10.4230/LIPIcs.SoCG.2022.58},
  annote =	{Keywords: multi-parameter persistent homology, stability, homology inference}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2021.27
Computing the Multicover Bifiltration

Authors: René Corbet, Michael Kerber, Michael Lesnick, and Georg Osang

Published in: LIPIcs, Volume 189, 37th International Symposium on Computational Geometry (SoCG 2021)

Abstract
Given a finite set A ⊂ ℝ^d, let Cov_{r,k} denote the set of all points within distance r to at least k points of A. Allowing r and k to vary, we obtain a 2-parameter family of spaces that grow larger when r increases or k decreases, called the multicover bifiltration. Motivated by the problem of computing the homology of this bifiltration, we introduce two closely related combinatorial bifiltrations, one polyhedral and the other simplicial, which are both topologically equivalent to the multicover bifiltration and far smaller than a Čech-based model considered in prior work of Sheehy. Our polyhedral construction is a bifiltration of the rhomboid tiling of Edelsbrunner and Osang, and can be efficiently computed using a variant of an algorithm given by these authors as well. Using an implementation for dimension 2 and 3, we provide experimental results. Our simplicial construction is useful for understanding the polyhedral construction and proving its correctness.
Cite as

René Corbet, Michael Kerber, Michael Lesnick, and Georg Osang. Computing the Multicover Bifiltration. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 27:1-27:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{corbet_et_al:LIPIcs.SoCG.2021.27,
  author =	{Corbet, Ren\'{e} and Kerber, Michael and Lesnick, Michael and Osang, Georg},
  title =	{{Computing the Multicover Bifiltration}},
  booktitle =	{37th International Symposium on Computational Geometry (SoCG 2021)},
  pages =	{27:1--27:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-184-9},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{189},
  editor =	{Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2021.27},
  URN =		{urn:nbn:de:0030-drops-138260},
  doi =		{10.4230/LIPIcs.SoCG.2021.27},
  annote =	{Keywords: Bifiltrations, nerves, higher-order Delaunay complexes, higher-order Voronoi diagrams, rhomboid tiling, multiparameter persistent homology, denoising}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2020.64
Fast Algorithms for Minimum Cycle Basis and Minimum Homology Basis

Authors: Abhishek Rathod

Published in: LIPIcs, Volume 164, 36th International Symposium on Computational Geometry (SoCG 2020)

Abstract
We study the problem of finding a minimum homology basis, that is, a shortest set of cycles that generates the 1-dimensional homology classes with ℤ₂ coefficients in a given simplicial complex K. This problem has been extensively studied in the last few years. For general complexes, the current best deterministic algorithm, by Dey et al. [Dey et al., 2018], runs in O(N^ω + N² g) time, where N denotes the number of simplices in K, g denotes the rank of the 1-homology group of K, and ω denotes the exponent of matrix multiplication. In this paper, we present two conceptually simple randomized algorithms that compute a minimum homology basis of a general simplicial complex K. The first algorithm runs in Õ(m^ω) time, where m denotes the number of edges in K, whereas the second algorithm runs in O(m^ω + N m^{ω-1}) time. We also study the problem of finding a minimum cycle basis in an undirected graph G with n vertices and m edges. The best known algorithm for this problem runs in O(m^ω) time. Our algorithm, which has a simpler high-level description, but is slightly more expensive, runs in Õ(m^ω) time.
Cite as

Abhishek Rathod. Fast Algorithms for Minimum Cycle Basis and Minimum Homology Basis. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 64:1-64:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{rathod:LIPIcs.SoCG.2020.64,
  author =	{Rathod, Abhishek},
  title =	{{Fast Algorithms for Minimum Cycle Basis and Minimum Homology Basis}},
  booktitle =	{36th International Symposium on Computational Geometry (SoCG 2020)},
  pages =	{64:1--64:11},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-143-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{164},
  editor =	{Cabello, Sergio and Chen, Danny Z.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.64},
  URN =		{urn:nbn:de:0030-drops-122223},
  doi =		{10.4230/LIPIcs.SoCG.2020.64},
  annote =	{Keywords: Computational topology, Minimum homology basis, Minimum cycle basis, Simplicial complexes, Matrix computations}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2019.37
Chunk Reduction for Multi-Parameter Persistent Homology

Authors: Ulderico Fugacci and Michael Kerber

Published in: LIPIcs, Volume 129, 35th International Symposium on Computational Geometry (SoCG 2019)

Abstract
The extension of persistent homology to multi-parameter setups is an algorithmic challenge. Since most computation tasks scale badly with the size of the input complex, an important pre-processing step consists of simplifying the input while maintaining the homological information. We present an algorithm that drastically reduces the size of an input. Our approach is an extension of the chunk algorithm for persistent homology (Bauer et al., Topological Methods in Data Analysis and Visualization III, 2014). We show that our construction produces the smallest multi-filtered chain complex among all the complexes quasi-isomorphic to the input, improving on the guarantees of previous work in the context of discrete Morse theory. Our algorithm also offers an immediate parallelization scheme in shared memory. Already its sequential version compares favorably with existing simplification schemes, as we show by experimental evaluation.
Cite as

Ulderico Fugacci and Michael Kerber. Chunk Reduction for Multi-Parameter Persistent Homology. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 37:1-37:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{fugacci_et_al:LIPIcs.SoCG.2019.37,
  author =	{Fugacci, Ulderico and Kerber, Michael},
  title =	{{Chunk Reduction for Multi-Parameter Persistent Homology}},
  booktitle =	{35th International Symposium on Computational Geometry (SoCG 2019)},
  pages =	{37:1--37:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-104-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{129},
  editor =	{Barequet, Gill and Wang, Yusu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.37},
  URN =		{urn:nbn:de:0030-drops-104413},
  doi =		{10.4230/LIPIcs.SoCG.2019.37},
  annote =	{Keywords: Multi-parameter persistent homology, Matrix reduction, Chain complexes}
}
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