CPW - Abstracts

Session 1, Monday (Sep 23)

13:00–13:20 CEST (7:00–7:20 EDT):

Tamal Krishna Dey, Michał Lipiński and Manuel Soriano-Trigueros . Conley-Morse barcodes – combinatorial continuation of a Morse decomposition.

Abstract: In recent years combinatorial dynamics has become an important subject of interest. The cornerstone for combinatorial modeling of continuous dynamics were combinatorial vector fields by Robin Forman. Building upon Forman’s work, Mrozek proposed the theory of combinatorial multivector fields, which was further generalized to general finite topological spaces, providing a comprehensive framework for combinatorial modeling of for continuous-time dynamical systems.

Central to the theory of combinatorial dynamical systems are isolated invariant sets (sets remaining invariant under the dynamics), the homological Conley index (an algebraic signature of an isolated invariant set capturing its dynamical nature) and Morse decomposition (a global summary of the gradient part of a dynamical system).

Recent research has utilized persistent homology to investigate various features of combinatorial dynamical systems, such as the robustness of an isolated invariant set and changes in the Conley index or Morse decomposition. Moreover, the language of persistent homology suits perfectly the concept of dynamical continuation which enables the study how an isolated invariant set and its Conley index evolve under continuous modifications of the underlying dynamical system such as a continuous change of the system parameter.

Our current work focuses on extending the concept of combinatorial continuation to the Morse decomposition. The extension will provide a summary of the changes of the global structure of the dynamical system as its parameter varies. Instead of studying a single isolated invariant set as previously we study all Morse sets (elements of Morse decomposition) simultaneously. A combinatorial perturbation of the system can result in a form of a combinatorial bifurcation. In consequence, Morse sets can aggregate together, disappear or change their structure. With the help of the developed theory of combinatorial continuation we can study the interplay of Conley indices of the bifurcating Morse sets.

The Conley index is defined through a relative homology of a pair of sets called an index pair. The relation between two Conley indices can be grasped through an appropriate comparison of index pairs. To see the global picture of how the Morse decomposition evolves we construct a web of inclusions between index pairs corresponding to Morse sets spanning an acyclic directed graph, which we call an index pairs transition diagram. By applying a homology functor we obtain a persistence module over a poset. However, as we show, unique properties of this dynamically inspired constructions guarantee that this persistence module admits a barcode de- composition. We call the outcome the Conley-Morse barcode. It comprehensively summarizes the exchange of the Conley index generators between the Morse sets and the global evolution of a combinatorial multivector field. A cartoon example of a parametrized dynamical system and the corresponding Conley-Morse barcode can be found in Figure 1. Figure 1: The top part shows a parametrized dynamical system on a sphere. Initially we can see a repelling critical point at the north pole and the attracting one on the south pole. As the parameter t increases the north pole goes through the Hopf bifurcation, the newly created periodic orbit moves downward, and eventually collapses into the south critical point. The bottom part shows the Conley-Morse barcode summarizing the evolution of the dynamical system. Each column represents Morse decomposition for a fixed parameter value, while each horizontal bar corresponds to a Conley index generator. The diagram shows how the generators are exchanged between consecutive Morse sets.

13:20-13:40 CEST (7:20–7:40 EDT)

Freya Jensen and Álvaro Torras-Casas. Persistent Homology Computation of Covered Alpha Complexes using the Mayer-Vietoris Spectral Sequence.

Abstract: Given a topological space and a cover by subspaces, the Mayer-Vietoris theorem relates the homology of the space with that of the cover elements and their intersections.
In this talk, I will introduce a new algorithm to compute the persistent Mayer--Vietoris spectral sequence in parallel for 2D and 3D alpha complexes.
The method starts from a disjoint cover of the underlying point cloud.
Next, I will describe how to compute in parallel the Alpha Complex from such a cover.
Afterwards, I will give an overview on how to compute the persistent Mayer--Vietoris spectral sequence from these covers.
One can recover persistent homology groups from such computation by solving the so-called extension problem.
This step I will explain with a simple but illustrative example.
Subsequently, I will present second-page collapse conditions that help speeding up the computations.
Finally, I will give an overview of an implementation in C++ using Open MPI and discuss some experimental results.

13:40-14:00 CEST (7:40–8:00 EDT)

Fabian Lenzen Cohomology methods in multi-parameter persistence.

Abstract: It is well-known that virtually all practical implementations of persistent homology rely on a certain duality between persistent homology and cohomology. Specifically, there is a 1:1-correspondence between the barcodes of the two. However, for Vietoris-Rips complexes, the usual column-reduction algorithm computes persistent cohomology much faster than homology; in particular if combined with certain optimization schemes such as clearing.

Similar dualities hold in any number of parameters, and in the two-parameter case, an algorithm relying on these has proven efficient. The real strength of this approach, however, lies in the fact that computing persistent cohomology allows for certain lazy evaluation schemes that are not applicable in the homology case. We expect that these evaluation schemes increase the efficiency of two-parameter persistent cohomology computation considerably.

Session 2, Monday (Sep 23)

14:30-15:00 CEST (8:30–9:00 EDT):

Invited Talk: Kathryn Hess-Bellwald MorphOMICs: a topological pipeline for microglia morphology analysis

Abstract: I will provide an overview of the MorphOMICs pipeline, a TDA tool for the analysis of highly complex and variable microglia morphology. I will explain some of the computational challenges faced in its development and utilization and present some of the results obtained via this pipeline.

(Joint work with G. Colombo, R. Cubero, L. Kanari, A. Venturino, R. Schulz, M. Scolamiero, J. Agerberg, H. Mathys, L.-H. Tsai, W. Chachólski, and S. Siegert)

15:00-15:20 CEST (9:00–9:20 EDT):

Thijs Beurskens , Tim Ophelders, Bettina Speckmann and Kevin Verbeek. An Interleaving Distance for Ordered Merge Trees.

Abstract: The interleaving distance is a common framework for comparing topological descriptors, including merge trees. Unfortunately, computing the interleaving distance between merge trees is known to be NP-hard. Instead, we consider ordered merge trees and we extend the interleaving distance to an order-preserving variant on this class of ordered merge trees. Due to the additional structure of our merge trees, we obtain an interleaving distance that is easy to compute. We build a theoretical foundation for this monotone interleaving distance, and show that it exhibits similar properties as
the regular interleaving distance. Furthermore, we establish a surprising connection between our new distance and the Fréchet distance for 1D curves.

15:20-15:40 CEST (9:20–9:40 EDT):

Philip Smith and Vitaliy Kurlin . Generic families of finite metric spaces with identical or trivial 1-dimensional persistence.

Abstract: This talk is based on the paper in APCT [8]. Topological Data Analysis (TDA) was pioneered by Herbert Edelsbrunner [4] and enjoyed important developments by Tamal Dey [3], Michael Kerber [5], Ulrich Bauer [1], Michael Lesnick [6], Elizabeth Munch [7] and others. The flagship tool of TDA is persistent homology summarizing geometric and topological features in unstructured data such as finite clouds of unordered points.

For many standard filtrations on point clouds such as Vietoris-Rips, Cech, and Delaunay, the resulting complexes and persistence are invariant under any isometry, which maintains all interpoint distances. Hence it makes sense to compare persistence with other isometry invariants by simplicity, speed, and strength, especially for classification tasks.

[8, Definition 3.2] describes how any point cloud in Rn can be extended by adding a generic tail of points without changing 1-dimensional persistence PD1, see figures at kurlin.org/research-papers.php.

[8, Corollary 4.5] gives sufficient conditions for a point cloud A in R^n to have the empty persistence PD1 and motivates the problem to find necessary conditions on A to have the empty persistence in all dimensions. We conjecture that all higher-dimensional persistence also remains unchanged by adding tails under the same conditions as in [8, Theorem 4.4].

Can we define an equivalence relation via geometric transformations such that the persistence (in all dimensions) is a complete invariant under this equivalence?

As an isometry invariant, persistence can be considered a partial solution to the problem with the stronger conditions (b,c), solved now by distance-based invariants [2, 9, 10].

Problem. Find a function I on clouds of unordered points satisfying these conditions:

(a) invariance: if any clouds A,B in R^n are related by rigid motion, then I(A) = I(B);

(b) completeness: if I(A) = I(B), the clouds A,B are related by rigid motion in R^n;

(c) reconstruction: any A in R^n can be reconstructed from I(A) uniquely under motion;

(d) Lipschitz continuity: there is a constant λ and a metric d satisfying all metric axioms on invariant values such that if a cloud B is obtained from another cloud A in R^n by perturbing every point of A up to Euclidean distance ε, then d(I(A), I(B)) ≤ λε.

(e) computability: for a fixed dimension n, the invariant I, the metric d, and a reconstruction of A in R^n can be obtained in a polynomial time in the size of a given cloud.

[1] Ulrich Bauer. Ripser: efficient computation of Vietoris–Rips persistence barcodes. Journal of Applied and Computational Topology, 5(3):391–423, 2021.

[2] Mireille Boutin and Gregor Kemper. On reconstructing n-point configurations from the distribution of distances or areas. Advances in Applied Mathematics, 32(4):709–735, 2004.

[3] Tamal K Dey and Cheng Xin. Generalized persistence algorithm for decomposing multiparameter persistence modules. J Applied and Computational Topology, 6(3):271–322, 2022.

[4] H Edelsbrunner, D Letscher, and A Zomorodian. Topological persistence and simplification. In Foundations of Computer Science, pages 454–463, 2000.

[5] Michael Kerber, Dmitriy Morozov, and Arnur Nigmetov. Geometry helps to compare persistence diagrams. ACM Journal of Experimental Algorithmics, 22:1–20, 2017.

[6] Michael Lesnick. The theory of the interleaving distance on multidimensional persistence modules. Foundations of Computational Mathematics, 15(3):613–650, 2015.

[7] J Perea, E Munch, and F Khasawneh. Approximating continuous functions on persistence diagrams using template functions. Foundations of Comp. Mathematics, 23:1215–1272, 2023.

[8] Philip Smith and Vitaliy Kurlin. Generic families of finite metric spaces with identical or trivial 1-dimensional persistence. Journal of Applied and Computational Topology, doi:10.1007/s41468-024-00177-6, 2024.

[9] Daniel Widdowson and Vitaliy Kurlin. Resolving the data ambiguity for periodic crystals. Advances in Neural Information Processing Systems (NeurIPS), 35:24625–24638, 2022.

[10] Daniel Widdowson and Vitaliy Kurlin. Recognizing rigid patterns of unlabeled point clouds by complete and continuous isometry invariants with no false negatives and no false positives. Proceedings of CVPR, pages 1275–1284, 2023.

15:40-16:00 CEST (9:40–10:00 EDT):

Gregory Henselman-Petrusek . Open Applied Topology: A fast, flexible, user-friendly tool for matrix algebra in TDA.

Abstract: Many problems in TDA have solutions in homological – and indeed, linear – algebra.
However, it’s challenging to harness these solutions, computationally. Matrices are often large, having millions or billions of rows and columns. They are indexed by simplices, rather than integers. They have coefficients in abstract fields and require exact numerical accuracy. They have unusual sparsity patterns.

This talk will introduce an open-source library to address some of these problems, Open Applied Topology (OAT).

OAT is a high-performance linear algebra solver with a user-friendly front end. It allows the user to perform mathematical operations including matrix/vector addition, multiplication, and factorization (R=DV, RU=D, U-match), and to compute persistence diagrams, barcodes, (optimal) (co)cycle representatives, and induced maps. Users can easily link the library to new types of chain complexes (simplicial, cubical, etc.), and to Python libraries such as SciPy. In sum, OAT is a user-friendly tool for matrix algebra in TDA.

If time remains, we will examine broader trends in TDA software development across the research landscape (government, private, academic), and opportunities for future growth and cooperation.

Session 3, Monday (Sep 23)

16:30-17:00 CEST (10:30–11:00 EDT):

Invited Talk: Magnus Botnan. How to prove hardness for Lp distances.

Abstract: In recent joint work with Håvard Bjerkevik, we show that computing the p-presentation distance is NP-hard both for merge trees and multiparameter persistence modules. While the technical details differ, the two proofs follow the same general strategy. In this talk, I will detail this strategy and outline the proof for the case of merge trees.

17:00-17:20 CEST (11:00–11:20 EDT):

Xinyi Wang , Elizabeth Munch and Carola Wenk. Computing the Bottleneck Distance from Every Direction.

Abstract: There are many ways to compare two persistence diagrams. One of the most common distances is the bottleneck distance. When the persistence diagrams of a shape in R^d are computed from every direction in S^{d−1}, we obtain the persistent homology transform (PHT). In this work, we develop a new kinetic data structure to compute the bottleneck distance between two PHTs obtained from shapes in R^2 from every direction. We provide the events and necessary updates to maintain the bottleneck distance between the diagrams using this structure. Our resulting algorithm runs in O(n2 log2 n). Furthermore, we show that this data structure is not limited to the directional transform setting since the techniques apply to more general vineyard structures.

17:20-17:40 CEST (11:20–11:40 EDT):

Soham Mukherjee, Shreyas Samaga , Cheng Xin, Steve Oudot and Tamal Dey. D-Gril: End-to-End Topological Learning with 2-parameter Persistence.

Abstract: End-to-end topological learning using 1-parameter persistence is well known. We show that the framework can be enhanced using 2-parameter persistence by adopting a recently introduced 2-parameter persistence based vectorization technique called GRIL. We establish a theoretical foundation of differentiating GRIL producing D-GRIL. We show that D-GRIL can be used to learn a bifiltration function on standard benchmark graph datasets. Further, we exhibit that this framework can be applied in the context of bio-activity prediction in drug discovery.

17:40-18:00 CEST (11:40–12:00 EDT):

Benjamin Jones and Guo-Wei Wei. Efficient Computation of Persistent Laplacian.

Abstract: We introduce a novel algorithm to aid in the computation of persistent topological Laplacians. We implement this algorithm and other existing persistent Laplacian algorithms in an efficient and open-source C++ library with Python bindings. As part of this library, we interface with several common complexes used in topological data analysis.

Session 4, Tuesday(Sep 24)

13:00–13:20 CEST (7:00–7:20 EDT):

Jan Jendrysiak , Michael Kerber and Tamal Dey. Decomposing Multiparameter Persistence Modules.

Abstract: ---

13:20-13:40 CEST (7:20–7:40 EDT):

Michael Bleher . Fast computation of pathwise persistence in pandemic-scale time series data of SARS-CoV-2 genomes.

Abstract: This talk showcases a methodology for the rapid computation of persistent homology in pandemic-scale time series data of SARS-CoV-2 genomes. By focusing only on small genomic distance scales, which capture the selective advantages and deleterious effects of individual mutations relative to its progenitor, and encoding temporal information through a deformation of the metric, we are able to efficiently calculate homology classes for millions of unique viral sequences. We discuss the implementation challenges and solutions that have enabled us to process data at this scale and explain the biological insights we obtain about the evolutionary fitness landscape during the pandemic. Our method exemplifies the application of topology to large datasets, streamlining data analysis at unprecedented scales, while also enhancing our understanding of viral evolution by providing insights into mutation dynamics at a daily resolution.

This is joint work with a large group of collaborators, including Maximilian Neumann, Mathieu Carriere, Ulrich Bauer, Raul Rabadan, and Andreas Ott and is based on

Bleher, M. _et al._ (2021) ‘Topological data analysis identifies emerging adaptive
mutations in SARS-CoV-2’. arxiv.org/abs/2106.07292.

Neumann, M. _et al._ (2022) ‘MuRiT: Efficient Computation of Pathwise Persistence Barcodes in Multi-Filtered Flag Complexes via Vietoris-Rips Transformations’. doi.org/10.48550/arXiv.2207.03394.

13:40-14:00 CEST (7:40–8:00 EDT)

Arnur Nigmetov and Dmitriy Morozov. Topological Optimization with Big Steps.

Abstract: Using persistent homology to guide optimization has emerged as a novel application of topological data
analysis. Existing methods treat persistence calculation as a black box and
backpropagate gradients only onto the simplices involved in particular
pairs. We show how the cycles and chains used in the persistence calculation
can be used to prescribe gradients to larger subsets of the domain.
In particular, we show that in a special case, which serves as a building
block for general losses, the problem can be solved exactly in linear time.
This relies on another contribution of this paper, which
eliminates the need to examine a factorial number of permutations of
simplices with the same value.
We present empirical experiments that show the practical
benefits of our algorithm: the number of steps required for the
optimization is reduced by an order of magnitude.

Session 5, Tuesday(Sep 24)

14:30-15:00 CEST (8:30–9:00 EDT):

Invited Talk: Heather Harrington . Topology of spatiotemporal trajectories.

Abstract: Many processes in the life sciences are inherently multi-scale and dynamic. Spatial structures and patterns vary across levels of organisation, from molecular to multi-cellular to multi-organism. With more sophisticated mechanistic models and data available, quantitative tools are needed to study their evolution in space and time. Topological data analysis (TDA) provides a multi-scale summary of data. We review persistent homology and present applications to cancer histology and spatial transcriptomics. Recent work by Kim and Memoli proposed a filtration for the case of dynamic metric spaces, requiring three parameter persistence. In-progress work by Lesnick, Bender and Gäfvert combines Groaner bases algorithms to compute minimal presentations of multiparameter persistence. Here we build on this work and present an algorithm, GBlandscapes, which computes 3-parameter persistent homology landscapes. We highlight its utility with concrete case studies of spatio-temporal trajectories arising in biological systems.

15:00-15:20 CEST (9:00–9:20 EDT):

Musashi Koyama , Facundo Mémoli, Vanessa Robins and Katharine Turner. Computing degree-1 Vietoris-Rips persistent homology more efficiently.

Abstract: Vietoris-Rips persistent homology is a widely used type of persistent homology to analyse the shape of point clouds. In particular, degree-1 Vietoris-Rips persistent homology is useful for detecting loop structures in space, but comes with the drawback of being computationally too expensive to apply to the large data sets encountered in the modern world.

Ripser is currently one of the most widely utilised options for computing degree-1 Vietoris-Rips persistent homology, but typically struggles with analyzing large point clouds due to memory limitations.

We present a modified version of the standard reduction algorithm for point clouds in Euclidean space and show the results for code optimized to compute degree-1 persistent homology for point clouds in 2 and 3 dimensional Euclidean space.

15:20-15:40 CEST (9:20–9:40 EDT):

Primoz Skraba and Amit Patel. Möbius Homology.

Abstract: This talk will introduce and discuss Mobius homology, a homology theory for representations of finite posets into abelian categories. While the connection between poset topology and Mobius functions is classical, we establish a direct connection between poset topology and Mobius inversions. More precisely, the Mobius homology categorifies the Mobius inversion because its Euler characteristic is equal to the Mobius inversion of the dimension function of the representation.

We also introduce a homological version of Rota's Galois Connection Theorem which relates the Mobius homology over two posets connected by a Galois connection. Our main application is to persistent homology over general posets. We show that under one definition, the persistence diagram is an Euler characteristic over a poset of intervals and hence Mobius homology is a categorification of the persistence diagram. This provides a new invariant for persistent homology over general posets. Finally, we use our homological Rota's Galois Connection Theorem to prove several results about the persistence diagram.

The full work paper be found at arxiv.org/abs/2307.01040

15:40-16:00 CEST (9:40–10:00 EDT):

Chunyin Siu , Gennady Samorodnitsky, Christina Lee Yu and Andrey Yao. Detection of small holes by the scale-invariant robust density-aware distance (RDAD) filtration.

Abstract: A novel topological-data-analytical (TDA) method is proposed to distinguish, from noise, small holes surrounded by high-density regions of a probability density function. The proposed method is robust against additive noise and outliers. Traditional TDA tools, like those based on the distance filtration, often struggle to distinguish small features from noise, because both have short persistences. An alternative filtration, called the Robust Density-Aware Distance filtration, is proposed to prolong the persistences of small holes of high-density regions. This is achieved by weighting the distance function by the density in the sense of Bell et al. The concept of distance-to-measure is incorporated to enhance stability and mitigate noise. The persistence-prolonging property and robustness of the proposed filtration are rigorously established, and numerical experiments are presented to demonstrate the proposed filtration’s utility in identifying small holes. This abstract is based on
doi.org/10.1007/s41468-024-00166-9
For a version without a paywall, see
link.springer.com/epdf/10.1007/s41468-024-00166-9

Session 6, Tuesday(Sep 24)

16:30-17:00 CEST (10:30–11:00 EDT):

Invited Talk: Tao Hou . Can zigzag persistence be computed as efficiently as the standard version?

Abstract: Zigzag persistence has proven a critical extension of the standard non-zigzag persistence. However, its computation is in general considered to be harder than the non-zigzag version. In this talk, I will review some of our recent efforts on bridging the efficiency gap between computing the zigzag and non-zigzag persistence. Specifically, I will talk about algorithms for the following problems: (1) computing the barcode from a zigzag filtration; (2) computing zigzag barcodes for graph filtrations; (3) updating zigzag persistence over local changes; (4) updating zigzag persistence for graph filtrations; (5) computing representatives for zigzag.

This is joint work with Tamal K. Dey, Dmitriy Morozov, and Salman Parsa.

17:00-17:20 CEST (11:00–11:20 EDT)

Minh Le . Persistent homology with k-nearest-neighbor filtrations reveals topological convergence of PageRank.

Abstract: Graph-based representations of point-cloud data are widely used in data science and machine learning, including \epsilon-graphs that contain edges between pairs of data points that are nearer than \epsilon and kNN-graphs that connect each point to its k-nearest neighbors. Recently, topological data analysis has emerged as a family of mathematical and computational techniques to investigate topological features of data using simplicial complexes. These are a higher-order generalization of graphs and many techniques such as Vietoris-Rips (VR) filtrations are also parameterized by a distance \epsilon. Here, we develop kNN complexes as a generalization of kNN graphs, leading to kNN-based persistent homology techniques for which we develop stability and convergence results. We apply this technique to characterize the convergence properties PageRank, highlighting how the perspective of discrete topology complements traditional geometrical-based analyses of convergence. Specifically, we show that convergence of relative positions (i.e., ranks) is captured by kNN persistent homology, whereas persistent homology with VR filtrations coincides with vector-norm convergence. In general, kNN-based persistent homology is expected to be useful to other data-science applications in which the relative positioning of data points is more important than their precise locations.

17:20-17:40 CEST (11:20–11:40 EDT):

Matthew Piekenbrock and Jose Perea. Scaling persistence using the Rank Invariant.

Abstract: Using a duality result between persistence diagrams and persistence measures, we introduce a framework for constructing families of continuous relaxations of the persistent rank invariant for parametrized families of persistence vector spaces indexed over the real line. Like the rank invariant, these families obey inclusion-exclusion, are derived from simplicial boundary operators, and encode all the information needed to construct a persistence diagram. Unlike the rank invariant, these spectrally-derived families enjoy a number of stability and continuity properties typically reserved for persistence diagrams, such as smoothness and differentiability over the positive semi-definite cone. Surprisingly, by leveraging a connection between stochastic Lanczos quadrature and implicit trace estimation, we show how to iteratively approximate all of the persistence invariants---Betti numbers, persistent pairs, and cycle representatives---in a linear space and "matrix-free" fashion amenable to GPU parallelization.

17:40-18:00 CEST (11:40–12:00 EDT):

Håvard Bakke Bjerkevik . Computational complexity in multiparameter persistence.

Abstract: For topological data analysis to be useful, we need efficient algorithms for computation. Our hunt for computational efficiency involves understanding which problems are NP-hard, and which problems might allow polynomial time algorithms.
It is known that computing the interleaving distance for merge trees and multiparameter modules is NP-hard, and in recent work, Magnus Botnan and I proved that NP-hardness also holds in these cases for $\ell^p$-versions of the interleaving distances. I will discuss these results and related classes of problems for which the computational complexity is still poorly understood. This includes an important meta-question in multiparameter persistence; namely, how much information do we have to sacrifice to turn a hard problem into a tractable one?

Session 7, Wednesday(Sep 25)

13:00–13:20 CEST (7:00–7:20 EDT):

Martina Flammer . Spatiotemporal Persistence Landscapes.

Abstract: We propose a method to apply and visualize persistent homology of time series. Our method captures persistent features in space and time in contrast to the existing procedures where one usually chooses one while keeping the other fixed. For this, we define an extended zigzag module that we build on a time-varying point cloud. This module combines ideas from zigzag persistent homology and multi-parameter persistent homology. Furthermore, we extend multi-parameter persistence landscapes to the case of extended zigzag modules using a recent generalization of the rank invariant by Kim and Memoli. To validate our proposed technique, we apply it to artificial as well as real world data.

13:20-13:40 CEST (7:20–7:40 EDT):

Michael Kerber and Florian Russold . Graphcode: Learning from multiparameter persistent homology using graph neural networks.

Abstract: We introduce graphcodes, a novel multi-scale summary of the topological properties of a dataset that is based on the well-established theory of persistent homology. Graphcodes handle datasets that are filtered along two real-valued scale parameters. Such multi-parameter topological summaries are usually based on complicated theoretical foundations and difficult to compute; in contrast, graphcodes yield an informative and interpretable summary and can be computed as efficient as one-parameter summaries. Moreover, a graphcode is simply an embedded graph and can therefore be readily integrated in machine learning pipelines using graph neural networks.

13:40-14:00 CEST (7:40–8:00 EDT):

Mikael Vejdemo-Johansson and Primoz Skraba. Copresentations of persistence modules.

Abstract: Traditionally the field of persistence modules has been focused on finite free (projective) presentations of modules - where a module is described as a cokernel of a map between projective modules. We find that for some tasks, notably when working with persistent cohomology, a copresentation describing a module as the kernel of a map between injective persistence modules is easier to work with.

In this talk we will describe the notion of copresentations, and demonstrate the most commonly used algebraic constructions with copresentations - including computing kernels, cokernels, images, pushouts and pullbacks, as well as methods to convert between presentations and copresentations. For all our constructions we have estimates of computational complexity for their use.

Session 8, Wednesday(Sep 25)

14:30-15:00 CEST (8:30–9:00 EDT):

Invited Talk: Marian Mrozek . A dynamic look at persistent homology: The depth poset of a filtered Lefschetz complex.

Abstract: The present-day methodology of persistent homology is dominated by the approach grounded in an algebraic decomposition theorem of persistence modules. However, the original approach by Edelsbrunner, Letscher, Zomorodian (2002) has a more geometric flavour. It is based on a birth-death pairing in an ordered sequence of cells of a simplicial complex called a filter. A filter may be interpreted as a Morse function inducing a combinatorial dynamical system on the simplicial complex. All birth-death pairs of the filter may be captured by a sequence of simplifications or smoothing of the system. The smoothing is achieved by a sequence of cancellations of shallow birth-death pairs. In dynamical terms, these shallow birth-death pairs are nearby critical cells joined by a heteroclinic connection. Originally, not all birth-death pairs have to be shallow and some cancellations are needed to make other birth-death pairs shallow. This leads to a poset structure in the collection of birth-death pairs. The induced dynamical system together with the poset, called the depth poset, strengthens the persistence diagram as a tool in TDA which may provide a better insight into the shape of data.

Based on a project in progress with Herbert Edelsbrunner. In collaboration with Micha? Lipi?ski and Manuel Soriano Trigueros.

15:00-15:20 CEST (9:00–9:20 EDT):

Amritendu Dhar , Vijay Natarajan and Abhishek Rathod. Geometric Localization of Homology Cycles.

Abstract: With the growth of TDA in applications, there is an
increasing need for computing homology cycles that
localize given homology classes or constitute a basis
for the homology group. Often applications require
these cycles to be tightest possible or geometry-aware
in some sense rather than being completely oblivious
of the embedding space. This demand has led to
studying homologous or basis cycles under various
optimization criteria. Computing an optimal cycle in a
given homology class, also referred to as the homology
localization problem, is an NP-hard problem in general.
Furthermore, there is currently no known optimality
criterion that localizes classes geometrically and admits
a stability property under the setting of persistent
homology. We present a geometric optimization of the
cycles that is computable in polynomial time and is
stable in an approximate sense.

15:20-15:40 CEST (9:20–9:40 EDT):

Alex Fernandes , Steve Oudot and François Petit. Computation of γ-linear projected barcodes for multiparameter persistence.

Abstract: The gamma-linear projected barcode was recently introduced as an alternative to the well-known fibered barcode for multiparameter persistence, in which restrictions of the modules to lines are replaced by pushforwards of the modules along linear forms in the polar of some fixed cone gamma. So far, the computation of the gamma-linear projected barcode has been studied only in the functional setting, in which persistence modules come from the persistent cohomology of vector fields. Here we develop a method that works in the algebraic setting directly, for any multiparameter persistence module over the euclidean space that is given via a finite free resolution. Our approach is similar to that of RIVET: first, it pre-processes the resolution to build an arrangement in the dual of the euclidean space and a barcode template in each face of the arrangement; second, given any query linear form u in the polar of gamma, it locates u within the arrangement to produce the corresponding barcode efficiently. While our theoretical complexity bounds are similar to the ones of RIVET, our arrangement turns out to be simpler thanks to the linear structure of the space of linear forms. Our theoretical analysis combines sheaf-theoretic and module-theoretic techniques, showing that multiparameter persistence modules can be converted into a special type of complexes of sheaves on vector spaces called conic-complexes, whose derived pushforwards by linear forms have predictable barcodes.

15:40-16:00 CEST (9:40–10:00 EDT):

Erin Wolf Chambers, Ishika Ghosh , Elizabeth Munch, Sarah Percival and Bei Wang. Computing a Loss Function to Bound the Interleaving Distance for Mapper Graphs.

Abstract: Mapper graphs preserve the connected components of the inverse image function f: X → R over any given cover. Inspired by the interleaving distance for Reeb graphs, (Chambers et al. 2024) extends this notion of distance to discretized mapper graphs. The distance is upper-bounded using a loss function. Unlike the NP-hard interleaving distance computation for Reeb graphs, the algorithm of the loss function has polynomial complexity. In this paper, we implement the categorical framework of mapper graphs and compute the loss function to bound the interleaving distance.

Session 9, Wednesday(Sep 25)

16:30-17:00 CEST (10:30–11:00 EDT):

Invited Talk: Yusu Wang. Enhancing graph learning models with topological information.

Abstract: Graph learning models (both message passing graph neural networks and graph transformers) face several challenges in effectively encoding large-scale graph structures within the learning model. In the last few years there have been various (potentially higher order) learning models to encoder richer information (e.g., simplicial or cellular message passing networks). In this talk, we will describe some of our recent work in this direction, including enhancing message passing neural networks with persistent homology, or with the cycle basis information, as well as enhancing transformer based graph learning models. This talk is based on several pieces of joint work with multiple collaborators which we will mention during the talk.

17:00-17:20 CEST (11:00–11:20 EDT):

Adam Onus , Primoz Skraba and Amit Patel. Local systems for periodic data.

Abstract: Periodic point clouds naturally arise when modelling large homogenous structures like crystals. They are naturally attributed with a map to the d-dimensional torus given by the quotient of translational symmetries, however there are many surprisingly subtle problems one encounters when studying their (persistent) homology. On one hand, the quotient space of a periodic data set loses information since it is a finite representation of an infinite object. On the other hand, because the quotient space exists in an ambient space whose homotopy type is that of the $d$-dimensional torus, they often introduce spurious "toroidal cycles" which do not represent the topology of the overlying periodic data set. The classification of toroidal cycles is a fundamental obstruction in creating a (persistent) homology theory of periodic data sets, and has been studied to some extent by Onus and Robins (2021), Onus and Skraba (2022), and by others (unpublished).

We present a new approach based on the relatively recent development of bisheaves. This, is a useful tool to study periodic data sets, as it allows us to unify several different approaches to study such spaces and completely characterize toroidal cycles. The theory of bisheaves and with them, persistent local systems was recently introduced by MacPherson and Patel (2021) as a method to study data with an attributed map to a manifold through the fibres of this map. The theory allows one to study the data locally, while also naturally being able to appeal to local systems of (co)sheaves to study the global behaviour of this data.

We introduce the first (persistent) homology theory for periodic data sets in Euclidean space through the use of (persistent) bisheaves and persistent local systems. By providing a complete description of toroidal classes, it provides an algorithm for determining the homology of finite windows of the periodic spaces. Furthermore, we provide the algorithms for computing not only bisheaves but also for computing the persistent local systems from them in a process called isobisheafification. We show how this, in addition to providing a unified mathematical framework, is explicitly algorithmic and is implementable.

Finally, we will illustrate the approach using special cases as examples, such as when the ambient quotient space is homotopic S^1 or T^2, i.e. the underlying translation group are Z or Z^2. Even in these simple cases, there is interesting additional structure which we expand upon.

Session 10, Thursday (Sep 26)

13:00–13:20 CEST (7:00–7:20 EDT):
Matías Bender , Oliver Gäfvert and Lesnick Michael. Computing minimal presentations of multi-parameter persistent homology.

Abstract: Within Topological data analysis, the area of Multi-parameter Persistent Homology (MPH) is a generalization of its single parameter analogue, persistent homology. MPH has a huge potential to analyse multi-modal data, time-varying data, and extract information robust to noise and outliers. However, efficient algorithms and publicly available implementations to compute MPH modules are essential missing pieces to make MPH useful in practice.
In this work, we develop new algorithms to compute the MPH module together with efficient C++ implementations collected in a new software package called Muphasa. Following [Carlsson et. al., 2010], it is based on Groebner basis computations. Our approach extends previous work on the 2-parameter case, and draws on ideas underlying the F4 and F5 algorithms for Groebner basis computation. In the r-parameter case, it computes a presentation for the homology of the chain complex of free modules A -> B -> C in O(r^2 * |B|^(r+1) + |B|^r * |C| + |B|^(r−1) * |C|^2 + r * |A| * |B|^2) arithmetic operations. We compare the efficiency of our approach against general-purpose strategies and show that ours is substantially faster and more memory efficient.

13:20-13:40 CEST (7:20–7:40 EDT):

Mikael Vejdemo-Johansson and Jordan Matuszewski. Symmetric Point Clouds and faster Vietoris-Rips generation.

Abstract: We study the computation of invariants of Vietoris-Rips complexes on point clouds with high degrees of known symmetry, and can show significant execution time and memory usage improvements on several examples. Our improvements use a local minimum of the action of the Cayley graph of a known presentation of the symmetry group as a first-line triage filter.

We use this faster Vietoris-Rips generation to compute Euler characteristics of Vietoris-Rips complexes on the hyper cubes and other examples of highly symmetric lattices and polytopes.

13:40-14:00 CEST (7:40–8:00 EDT):

Abhinav Natarajan , Thomas Chaplin, Adam Brown and Maria-Jose Jimenez. Morse Theory for Chromatic Delaunay Triangulations.

Abstract: The chromatic alpha filtration is a generalization of the alpha filtration that can encode spatial relationships among classes of labelled point cloud data, and has applications in topological data analysis of multi-class data. In our work we introduce the chromatic Delaunay–Cech and chromatic Delaunay–Rips filtrations, which are computationally favourable alternatives to the chromatic alpha filtration. We use generalized discrete Morse theory to show that the Cech, chromatic Delaunay-Cech, and chromatic alpha filtrations are related by simplicial collapses. Our result generalizes a result of Bauer and Edelsbrunner from the non-chromatic to the chromatic setting, and shows that the chromatic Delaunay-Cech filtration can be used as a drop-in alternative to the chromatic alpha filtration for persistent homology computations.

We also show that the chromatic Delaunay–Rips filtration is locally stable to perturbations of the underlying point cloud. This local stability, in conjunction with our previous result, means that the chromatic Delaunay-Rips filtration is a viable approximation to the chromatic alpha filtration for persistent homology calculations, with the advantage of being much faster to compute.

In this talk I will give a sketch of the proofs of our main results, and elaborate on how these results provide theoretical justification for the use of chromatic Delaunay–Cech and chromatic Delaunay–Rips filtrations in practical applications. I will also show the data from numerical experiments that demonstrate the computational advantage of our constructions.

Session 11, Thursday (Sep 26)

14:30-15:00 CEST (8:30–9:00 EDT):

Invited Talk: Steve Oudot . Persistent Homology of Multiscalar Fields from Point Cloud Data.

Abstract: Given a metric space X and a function f:X->R^d, both unknown, can we reliably estimate the persistent homology of f from the sole knowledge of the pairwise distances and function values at a finite collection of points sampled from X? This problem was addressed in the case d=1 by Chazal and co-authors in a 2009 paper that then served as the foundation for the ToMATo clustering algorithm. Here we generalize the approach to arbitrary d, connecting it to the function-Rips multifilration and lifting some of the limitations of the original approach along the way. This is joint work with Ethan André and Jingyi Li.

15:00-15:20 CEST (9:00–9:20 EDT):

Álvaro Torras-Casas, Eduardo Paluzo-Hidalgo and Rocío Gonzalez-Diaz. Towards topological data quality.

Abstract: This talk introduces a novel method for assessing data quality using topological data analysis techniques. We propose a new topological invariant derived from persistence matchings specifically designed for subsets of finite metric spaces. Our approach includes an algorithm that computes this invariant using minimum spanning trees. By relating the invariant to the images, kernels, and cokernels of persistence morphisms induced by inclusions, we can qualitatively analyze how well a subset captures the cluster structure of a larger dataset. Additionally, our method estimates bounds for the Hausdorff distance between a subset and the full dataset. Finally, we propose applying this methodology to explain why a particular subset might lead to poor performance in a supervised learning model.

15:20-15:40 CEST (9:20–9:40 EDT)):

Charles Arnal , Vincent Divol and David Cohen-Steiner. Critical points and sampling theory of generic submanifolds.

Abstract: We are interested in compact submanifolds and in random finite samplings of those submanifolds, in the distance functions to the submanifolds and to their samplings and in the critical points and the persistence diagrams of those distance functions.
These can be very irregular in general, but we show that they are well-behaved when the submanifold is generic.

15:40-16:00 CEST (9:40–10:00 EDT):

Ran Levi, Jacek Brodzki and Henri Riihimäki . Differential calculus for modules over posets.

Abstract: The motivation for this project arises from developing a new type of analysis for persistence modules. In its general form one defines persistence modules as functors from an arbitrary poset P to some abelian target category. In other words, a persistence module is simply a representation of the source poset, or a P-module. As such much research was dedicated to studying persistence modules in this context. However, when the source category is more general than a linear order, obtaining equivalents of barcodes poses challenges. In particular, keeping in mind that persistence is supposed to be applicable, computability sets limits to general persistence in applications. This talk will describe a new set of ideas for local analysis of modules over posets by methods borrowed from multivariable calculus on graphs. We will outline the mathematical steps behind developing a gradient module of a given P-module that captures infinitesimal changes in the module. Due to the local nature the gradient calculus is not restricted by the size nor shape of the indexing poset. We will demonstrate applications to commutative ladders and hierarchical clustering modules.

Session 12, Thursday (Sep 26)

16:30-17:00 CEST (10:30–11:00 EDT):

Invited Talk: Anastasios Stefanou . Combinatorial Topological Models for Phylogenetic Networks and the Mergegram Invariant

Abstract: Mutations of genetic sequences are often accompanied by their recombinations, known as phylogenetic networks. These networks are typically reconstructed from coalescent processes that may arise from optimal merging or fitting together a given set of phylogenetic trees. Nakhleh formulated the phylogenetic network reconstruction problem (PNRP): Given a family of phylogenetic trees over a common set of taxa, is there a unique minimal phylogenetic network whose set of spanning trees contains the family?

Inspired by ideas from topological data analysis (TDA), we devise lattice-diagram models for phylogenetic networks and filtrations, the cliquegram and the facegram, both generalizing the dendrogram (filtered partition) model of phylogenetic trees. Both models allow us to solve the PNRP rigorously. The solutions are obtained by taking the join of the dendrograms on the lattice of cliquegrams or facegrams. Furthermore, computing the join-facegram is polynomial in the size and number of the input trees.

Cliquegrams and facegrams can be challenging to work with when the number of taxa is large. We propose a topological invariant of facegrams and filtrations, called the mergegram, by extending a construction by Elkin and Kurlin defined on dendrograms. We show that the mergegram is invariant of weak equivalences of filtrations which, in turn, implies that it is a 1-Lipschitz stable invariant with respect to Mémoli's tripod distance. The mergegram, can be used as a computable proxy for phylogenetic networks and filtrations of datasets. We illustrate the utility of these new TDA-concepts to phylogenetics, by performing experiments with artificial and biological data.

This is joint work with Paweł Dłotko and Jan Felix Senge.

17:00-17:20 CEST (11:00–11:20 EDT):

Tim Ophelders, Anna Schenfisch , Willem Sonke and Bettina Speckmann. River and Intertidal Network Extraction via Volume Filtrations.

Abstract: Most models used to extract the network of water flow from terrain height assume that water follows the path of steepest descent. While this assumption is reasonable for steep terrains, it does not allow for the network to form branching or braided structures, which are common in nearly flat terrains. In our methods for network extraction in such settings, we begin by defining a discrete Morse function on a cubulated terrain. We form an initial network based on the ascending paths of this function, and then filter the network by a volume parameter. Conceptually, we remove features that would disappear if only a small volume of the terrain were altered. We first discuss our framework for river networks, and then discuss the more involved setting of intertidal networks.

17:20-17:40 CEST (11:20–11:40 EDT):

Barbara Giunti , Wojciech Chacholski, Claudia Landi and Francesca Tombari. Constructive projective approximations.

Abstract: Generally, objects in the persistence pipeline have two main dimensions, both crucial for obtaining a faithful, meaningful description of the data: the one given by the parametrizing poset, and the one given by the degrees of the (co)chain complex. In this work, we study the relationship between parametrizations and chain complexes, with the goal of defining constructible invariants.

The parametrization is given by continuous posets, since they allow for studying stability and being able to compare objects with features at different times. However, to have explicit constructions, we need some finiteness notion. In this work, we use the one of tameness. The invariants we focus on here are the minimal projective covers. We prove the existence of these invariants showing that, starting with a category C with minimal projective covers, they still exist when we take the chain complexes in C and when we tame parametrize the objects in C, under reasonable hypotheses. In both these results, the existence is proved with explicit constructions extending the minimal projective covers in C. Moreover, we can combine
these two operations and the order in which we do it does not matter for finite inputs.
This is particularly useful for computations, as it allows us to build minimal projective covers more efficiently.
These results are especially effective if C is the category of finite dimensional vector spaces, where every object is projective and thus the computation of minimal projective covers is trivial.

17:40-18:00 CEST (11:40–12:00 EDT):

Ángel Javier Alonso . Delaunay bifiltrations of functions on point clouds.

Abstract: Joint work with Michael Kerber, Tung Lam, Michael Lesnick.

The Delaunay filtration D(X) of a point cloud X is a central tool of computational topology. Its use is justified by the topological equivalence of D(X) and the offset (i.e., union-of-balls) filtration of X. Given a function γ:X -> R, we introduce a Delaunay bifiltration DC(γ) that satisfies an analogous topological equivalence, ensuring that DC(γ) topologically encodes the offset filtrations of all sublevel sets of γ, as well as the topological relations between them. DC(γ) is of size O(|X|^⌈(d+1)/2⌉), which for d odd matches the worst-case size of D(X). Adapting the Bowyer-Watson algorithm for computing Delaunay triangulations, we give a simple, practical algorithm to compute DC(γ) in time O(|X|^(⌈d/2⌉+1)). Our implementation, based on CGAL, computes DC(γ) with modest overhead compared to computing D(X), and handles tens of thousands of points in R^3 within seconds.
I will also talk about an extension to functions γ:X -> R^2 that is joint work with the same people plus Abhishek Rathod.

Session 13, Friday (Sep 27)

13:00–13:20 CEST (7:00–7:20 EDT):

Isaac Ren . Bar-to-bar morphisms and Wasserstein distances.

Abstract: Among morphisms between one-parameter persistence modules, we can isolate the so-called bar-to-bar morphisms: direct sums of morphisms from a single bar to a single bar. These bar-to-bar morphisms are simpler to study than arbitrary morphisms and, as we will see in this talk, they play a key role in the computation of some persistence invariants.

Our central result is the following: given an arbitrary monomorphism between two persistence modules, there exists a bar-to-bar monomorphism, between the same modules, whose cokernel has a smaller p-norm. We outline the proof of this result, and then present some of its consequences. Firstly, the monomorphism induced by the canonical matching of Bauer and Lesnick (2015) minimizes cokernel norm among all monomorphisms. Secondly, p-norms for persistence modules satisfy a subadditivity property for short exact sequences, which leads to the triangular inequality for Wasserstein distances. Lastly, we explicitly compute Wasserstein stable ranks, which are invariants of persistence modules defined using Wasserstein distances. Efficient algorithms for computing Wasserstein stable ranks and distances between them allow us to use these invariants in data analysis in combination with machine learning techniques.

This is a joint work with Jens Agerberg, Andrea Guidolin, and Martina Scolamiero. Our preprint is available here: arxiv.org/abs/2301.06484.

13:20-13:40 CEST (7:20–7:40 EDT):

Jens Kristian Refsgaard Schou and Bei Wang. PersiSort: A New Perspective on Adaptive Sorting Based on Persistence.

Abstract: Adaptive sorting exploits the structure of a partially sorted list---in particular, the sorted segments of a list called runs---to improve its performance.
Persistent homology, on the other hand, is a topological data analysis tool that captures a space's topological features at different scales.
In this paper, we combine these two seemingly unrelated concepts and introduce a new perspective on adaptive sorting.
We introduce a new stable sorting algorithm, referred to as the Persistence Sort (or PersiSort in short), which utilizes the persistence pairs among the local extrema of a list.
Given a list of n elements containing r runs with run entropy H, we prove, for the first time, that any adaptive sorting algorithm that uses the two-way-merge subroutine (AdaptMerge) of Carlsson et al (1990) performs O(nH)= O(n\log r) comparisons to merge precomputed runs based on its predetermined merge policy, and is therefore worst-case optimal.
Using PersiSort, we then provide a new way to analyze adaptive sorting with a persistence-based arrangement of runs.
Unlike previous work such as PowerSort and TimSort, PersiSort does not consider the number of elements in each run but the values of elements in the sorting process.
We finally discuss the scenarios when PersiSort outperforms several state-of-the-art adaptive sorting algorithms.

13:40-14:00 CEST (7:40–8:00 EDT):

Matteo Pegoraro . Data Analysis with Merge Trees.

Abstract: A merge tree (MT) is a topological summary encoding the merging pattern of path-connected components along a filtration of topological spaces. Such objects arise naturally in different scientific fields and fit very well into the framework of Topological Data Analysis. Compared to persistence diagrams (PDs), MTs provide a finer representation of such filtrations, making them an ideal alternative to PDs in some situations. On the other hand, MTs are computationally very expensive and this limits their use in applications. In this talk I will present a metric for MTs which I defined and which can be computed using linear integer programming. This metric satisfies some stability properties, allowing also for some geometric and topological investigations concerning geodesics and compact sets. Thanks to such results the existence of Frechét means is assessed and an approximation algorithm for such objects is proposed. Lastly, I will present an application in the field of functional data analysis and one in the field of medical imaging (radiomics).

Session 14, Friday (Sep 27)

14:30-15:00 CEST (8:30–9:00 EDT):

Invited Talk: Clement Maria. Algorithmic complexity in quantum topology.

Abstract: Quantum topology offers tools to construct topological invariants of knots and 3-manifolds, that are computable with simple exponential algorithms. On the one hand, we are given algebraic data (a fusion category) and on the other hand a topological object (we will focus on 3-manifolds presented by triangulations).

Quantum invariants are particularly powerful at discriminating non-homeomorphic 3-manifolds, and would be a key ingredient in the recognition of 3-manifolds that one could hope for in topological data analysis. However, the exponential nature of the construction prevents naive algorithms from running at the large scales of TDA.

In this talk, I will review some aspects of 3-manifold topology and the construction of 3-manifold quantum invariants. I will finally give an overview of their computational complexity, depending on both the fusion category and the input triangulated 3-manifold, and show that some situations are tractable.

15:00-15:20 CEST (9:00–9:20 EDT):

Barbara Giunti, Guillaume Houry, Michael Kerber and Matthias Soels . Expected Complexity of Persistent Homology Computation via Matrix Reduction.

Abstract: ---

15:20-15:40 CEST (9:20–9:40 EDT):

Hubert Wagner, Nickolas Arustamyan, Matthew Wheeler and Peter Bubenik. From Shape to Interactions: Quantifying Geometric-Topological Interactions using Mixup Barcodes.

Abstract: Studying the shape of data has proven to be useful in many applications.
In some cases, however, it is useful to determine the geometric-topological interactions between two or more subsets of the dataset. Motivated by such
situations we introduce mixup barcodes: an extension of persistence barcodes that additionally capture various interaction between subsets of data. In the talk,
we will focus on the intuition and computational aspects and briefly discuss applied aspects of this work.

15:40-16:00 CEST (9:40–10:00 EDT):

Benedikt Fluhr . Vectorizing Persistence Using Relative Homology.

Abstract: We propose a new method of vectorizing persistence diagrams based on the relative homology lattice introduced by Deheuvels. More specifically, we embed persistence diagrams into the Hilbert space of square-integrable functions on a 2-dimensional domain. This embedding has a straightforward interpretation in terms of the relative homology groups stemming from the offset filtration of the corresponding point cloud. Then we describe how we can use the dedicated library persunraveltorch to train support vector classifiers as well as a certain type of group equivariant convolutional neural network.