arXiv subjects

Ulrich Bauer

Publications and source records attributed to Ulrich Bauer.

At least 19 recordsLinked to original sources

arXiv2026

An Algebraic Introduction to Persistence

We introduce persistence with an emphasis on its algebraic foundations, using the representation theory of posets. Linear representations of posets arise in several areas of mathematics, including the representation theory of quivers and finite dimensional algebras, Morse theory and other areas of geometry, as well as topological inference and topological data analysis -- often via persistent homology. In some of these contexts, the category of poset representations of interest admits a metric structure given by the so-called interleaving distance. Persistence studies the algebraic properties of these poset representations and their behavior under perturbations in the interleaving distance. We survey fundamental results in the area, applications to pure and applied mathematics, advanced topics such as multiparameter persistence, as well as theoretical challenges and open questions.

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arXiv2026

Dualities in Multiparameter Persistence

In the theory of persistent homology, a well known duality relates the barcodes of the absolute homology and relative cohomology of a one-parameter simplicial filtration. Motivated by the problem of computing free presentations of the (co)homology of multiparameter Rips filtrations, we give a multiparameter generalization of this duality. Considering two duality functors on multiparameter persistence modules, the pointwise dual $(-)^*$ and the global dual $(-)^\dagger$, we show that $H_q(C)^* \cong H^{N+q}(C^\dagger)$ for chain complexes $C$ of free $N$-parameter persistence modules with acyclic colimit. We give an elementary and accessible proof based on a long exact sequence argument, and also give an alternate proof that casts the result as a special case of multigraded Grothendieck local duality. As a corollary, we recover a simple correspondence between minimal free resolutions of a persistence module $M$ and those of its pointwise dual $M^*$, a result previously obtained by Miller, 2000. These results form the foundation of a state-of-the-art algorithm for computing free resolutions of the homology of Vietoris--Rips bifiltrations, described in a forthcoming paper.

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arXiv2026

A metrically complete and Krull--Schmidt space of multiparameter persistence modules

We show that the observable category of q-tame multiparameter persistence modules satisfies good metric and algebraic properties: it forms a complete metric space with respect to the interleaving distance, and it is Krull--Schmidt in the sense that every object admits an essentially unique decomposition into indecomposables. Moreover, we show that these metric and algebraic structures are compatible: two objects are at distance zero if and only if they are isomorphic. We argue that the observable category of q-tame multiparameter persistence modules is the right setup for multiparameter persistence by showing that many of the categories already considered in the literature form full subcategory of this category. We also characterize precompact sets in terms of finite representation type of certain discretizations, and show that the image of several of the main constructions in multiparameter persistence is precompact.

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arXiv2025

Fast free resolutions of bifiltered chain complexes

In a $k$-critical bifiltration, every simplex enters along a staircase with at most $k$ steps. Examples with $k>1$ include degree-Rips bifiltrations and models of the multicover bifiltration. We consider the problem of converting a $k$-critical bifiltration into a $1$-critical (i.e. free) chain complex with equivalent homology. This is known as computing a free resolution of the underlying chain complex and is a first step toward post-processing such bifiltrations. We present two algorithms. The first one computes free resolutions corresponding to path graphs and assembles them to a chain complex by computing additional maps. The simple combinatorial structure of path graphs leads to good performance in practice, as demonstrated by extensive experiments. However, its worst-case bound is quadratic in the input size because long paths might yield dense boundary matrices in the output. Our second algorithm replaces the simplex-wise path graphs with ones that maintain short paths which leads to almost linear runtime and output size. We demonstrate that pre-computing a free resolution speeds up the task of computing a minimal presentation of the homology of a $k$-critical bifiltration in a fixed dimension. Furthermore, our findings show that a chain complex that is minimal in terms of generators can be asymptotically larger than the non-minimal output complex of our second algorithm in terms of description size.

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arXiv2025

Inferring a Cell Structure on the Space of Cyclooctane Conformations

The conformation space of cyclooctane, a ringlike organic molecule comprising eight carbon atoms, is a two-dimensional algebraic variety, which has been studied extensively for more than 90 years. We propose a cell structure representing this space, which arises naturally by partitioning the space into subsets of conformations that admit particular symmetries. We do so both for the labeled conformation space, in which the carbon atoms are considered as distinct, and for the actual, unlabeled, conformation space. The proposed cell structure is obtained by identifying subspaces of conformations based on symmetry patterns and studying the geometry and topology of these subsets using methods from dimensionality reduction and topological data analysis. Our findings suggest that, in contrast to the labeled variant, the conformation space of cyclooctane is contractible.

math.GN↗

arXiv2025

On the additive image of 0th persistent homology

For $X$ a finite category and $F$ a finite field, we study the additive image of the functor $\operatorname{H}_0(-,F) \colon \operatorname{rep}(X, \mathbf{Top}) \to \operatorname{rep}(X, \mathbf{Vect}_F)$, or equivalently, of the free functor $\operatorname{rep}(X, \mathbf{Set}) \to \operatorname{rep}(X, \mathbf{Vect}_F)$. We characterize all finite categories $X$ for which the indecomposables in the additive image coincide with the indecomposable indicator representations and provide examples of quivers of wild representation type where the additive image contains only finitely many indecomposables. Motivated by questions in topological data analysis, we conduct a detailed analysis of the additive image for finite grids. In particular, we show that for grids of infinite representation type, there exist infinitely many indecomposables both within and outside the additive image. We develop an algorithm for determining if a representation of a finite category is in the additive image. In addition, we investigate conditions for realizability and the effect of modifications of the source category and the underlying field. The paper concludes with a discussion of the additive image of $\operatorname{H}_n(-,F)$ for an arbitrary field $F$, extending previous work for prime fields.

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arXiv2024

Generating Cofaces in Vietoris--Rips Filtration Order

Cofaces -- simplices that contain a given simplex -- have multiple important uses in generating and using a Vietoris-Rips filtration: both in creating the coboundary matrix for computing persistent cohomology, and for generating the ordered sequence of simplices in the first place. Traditionally, most methods have generated simplices first, and then sorted them in filtration order after the generation step. In this paper, we propose fast algorithms for generating the sequence of simplices by generating cofaces of a given simplex with the same diameter, which by construction produces simplices in filtration order, and for generating additional cofaces in filtration order using sorted neighborhood lists in order to generate coboundaries directly in filtration order.

cs.CG↗

arXiv2024

Topograph: An efficient Graph-Based Framework for Strictly Topology Preserving Image Segmentation

Topological correctness plays a critical role in many image segmentation tasks, yet most networks are trained using pixel-wise loss functions, such as Dice, neglecting topological accuracy. Existing topology-aware methods often lack robust topological guarantees, are limited to specific use cases, or impose high computational costs. In this work, we propose a novel, graph-based framework for topologically accurate image segmentation that is both computationally efficient and generally applicable. Our method constructs a component graph that fully encodes the topological information of both the prediction and ground truth, allowing us to efficiently identify topologically critical regions and aggregate a loss based on local neighborhood information. Furthermore, we introduce a strict topological metric capturing the homotopy equivalence between the union and intersection of prediction-label pairs. We formally prove the topological guarantees of our approach and empirically validate its effectiveness on binary and multi-class datasets. Our loss demonstrates state-of-the-art performance with up to fivefold faster loss computation compared to persistent homology methods.

cs.CV↗

arXiv2024

Efficient Betti Matching Enables Topology-Aware 3D Segmentation via Persistent Homology

In this work, we propose an efficient algorithm for the calculation of the Betti matching, which can be used as a loss function to train topology aware segmentation networks. Betti matching loss builds on techniques from topological data analysis, specifically persistent homology. A major challenge is the computational cost of computing persistence barcodes. In response to this challenge, we propose a new, highly optimized implementation of Betti matching, implemented in C++ together with a python interface, which achieves significant speedups compared to the state-of-the-art implementation Cubical Ripser. We use Betti matching 3D to train segmentation networks with the Betti matching loss and demonstrate improved topological correctness of predicted segmentations across several datasets. The source code is available at https://github.com/nstucki/Betti-Matching-3D.

math.AT↗

arXiv2024

Topologically Faithful Multi-class Segmentation in Medical Images

Topological accuracy in medical image segmentation is a highly important property for downstream applications such as network analysis and flow modeling in vessels or cell counting. Recently, significant methodological advancements have brought well-founded concepts from algebraic topology to binary segmentation. However, these approaches have been underexplored in multi-class segmentation scenarios, where topological errors are common. We propose a general loss function for topologically faithful multi-class segmentation extending the recent Betti matching concept, which is based on induced matchings of persistence barcodes. We project the N-class segmentation problem to N single-class segmentation tasks, which allows us to use 1-parameter persistent homology, making training of neural networks computationally feasible. We validate our method on a comprehensive set of four medical datasets with highly variant topological characteristics. Our loss formulation significantly enhances topological correctness in cardiac, cell, artery-vein, and Circle of Willis segmentation.

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arXiv2023

Cycling Signatures: Identifying Cycling Motions in Time Series using Algebraic Topology

Recurrence is a fundamental characteristic of dynamical systems with complicated behavior. Understanding the inner structure of recurrence is challenging, especially if the system has many degrees of freedom and is subject to noise. We develop algebraic topological notions for identifying and classifying elementary recurrent motions -- called cycling -- and the transitions between those. Statistics on these cycling motions can be computed from sampled trajectories (time series data), providing coarse global information on the structure of the recurrent behavior. We demonstrate this through three examples; in particular, we identify and analyze six cycling motions in a four dimensional system with a hyperchaotic attractor. We see this as a promising approach to reveal coarse-grained dynamical information on high-dimensional systems.

math.DS↗

arXiv2023

Efficient two-parameter persistence computation via cohomology

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.

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arXiv2022

Wrapping Cycles in Delaunay Complexes: Bridging Persistent Homology and Discrete Morse Theory

We study the connection between discrete Morse theory and persistent homology in the context of shape reconstruction methods. Specifically, we consider the construction of Wrap complexes, introduced by Edelsbrunner as a subcomplex of the Delaunay complex, and the construction of lexicographic optimal homologous cycles, also considered by Cohen-Steiner, Lieutier, and Vuillamy in a similar setting. We show that for any cycle in a Delaunay complex for a given radius parameter, the lexicographically optimal homologous cycle is supported on the Wrap complex for the same parameter, thereby establishing a close connection between the two methods. We obtain this result by establishing a fundamental connection between reduction of cycles in the computation of persistent homology and gradient flows in the algebraic generalization of discrete Morse theory.

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arXiv2022

Topologically faithful image segmentation via induced matching of persistence barcodes

Image segmentation is a largely researched field where neural networks find vast applications in many facets of technology. Some of the most popular approaches to train segmentation networks employ loss functions optimizing pixel-overlap, an objective that is insufficient for many segmentation tasks. In recent years, their limitations fueled a growing interest in topology-aware methods, which aim to recover the correct topology of the segmented structures. However, so far, none of the existing approaches achieve a spatially correct matching between the topological features of ground truth and prediction. In this work, we propose the first topologically and feature-wise accurate metric and loss function for supervised image segmentation, which we term Betti matching. We show how induced matchings guarantee the spatially correct matching between barcodes in a segmentation setting. Furthermore, we propose an efficient algorithm to compute the Betti matching of images. We show that the Betti matching error is an interpretable metric to evaluate the topological correctness of segmentations, which is more sensitive than the well-established Betti number error. Moreover, the differentiability of the Betti matching loss enables its use as a loss function. It improves the topological performance of segmentation networks across six diverse datasets while preserving the volumetric performance. Our code is available in https://github.com/nstucki/Betti-matching.

cs.CV↗

arXiv2022

Multi-Parameter Persistence Modules are Generically Indecomposable

Algebraic persistence studies persistence modules (typically, linear representations of the poset $\mathbf{R}^n$ with $n \geq 1$) and the algebraic relationships between persistence modules that are interleaved. The notion of $\varepsilon$-interleaving between persistence modules is a generalization of the notion of isomorphism (recovering isomorphism when $\varepsilon = 0$), which can be used to quantify how far any two persistence modules are from being isomorphic. An emblematic example of this kind of study is the algebraic stability theorem, which strengthens the Krull--Schmidt property of one-parameter persistence modules (representations of $\mathbf{R}$) by generalizing isomorphism to interleaving: If a pair of one-parameter persistence modules is $\varepsilon$-interleaved, then there exists a partial matching between the indecomposable summands of the two modules such that matched indecomposables are $\varepsilon$-interleaved and unmatched indecomposables are $\varepsilon$-interleaved with the zero module. Our first main result implies that the obvious extension of the algebraic stability theorem to the case of multi-parameter persistence modules (representations of $\mathbf{R}^n$ with $n \geq 2$) fails spectacularly: Any finitely presentable multi-parameter persistence module can be approximated arbitrarily well by an indecomposable module. Our second main result states that modules that are sufficiently close to an indecomposable decompose as a direct sum of an indecomposable and a nearly trivial module. We derive from these two results several consequences about the interplay between the algebraic and the topological properties of multi-parameter persistence modules. These results provide strong motivation for approaching multi-parameter persistence in a way that does not rely on directly decomposing modules by indecomposables.

math.RT↗

arXiv2022

Keeping it sparse: Computing Persistent Homology revisited

In this work, we study several variants of matrix reduction via Gaussian elimination that try to keep the reduced matrix sparse. The motivation comes from the growing field of topological data analysis where matrix reduction is the major subroutine to compute barcodes, the main invariant therein. We propose two novel variants of the standard algorithm, called swap and retrospective reductions. We test them on a large collection of data against other known variants to compare their efficiency, and we find that sometimes they provide a considerable speed-up. We also present novel output-sensitive bounds for the retrospective variant which better explain the discrepancy between the cubic worst-case complexity bound and the almost linear practical behavior of matrix reduction. Finally, we provide several constructions on which one of the variants performs strictly better than the others.

cs.CG↗

arXiv2022

Relative Interlevel Set Cohomology Categorifies Extended Persistence Diagrams

The extended persistence diagram introduced by Cohen-Steiner, Edelsbrunner, and Harer is an invariant of real-valued continuous functions, which are $\mathbb{F}$-tame in the sense that all open interlevel sets have degree-wise finite-dimensional cohomology with coefficients in a fixed field $\mathbb{F}$. We show that relative interlevel set cohomology (RISC), which is based on the Mayer--Vietoris pyramid by Carlsson, de Silva, and Morozov, categorifies this invariant. More specifically, we define an abelian Frobenius category $\mathrm{pres}(\mathcal{J})$ of presheaves, which are presentable in a certain sense, such that the RISC $h(f)$ of an $\mathbb{F}$-tame function $f \colon X \rightarrow \mathbb{R}$ is an object of $\mathrm{pres}(\mathcal{J})$, and moreover the extended persistence diagram of $f$ uniquely determines - and is determined by - the corresponding element $[h(f)] \in K_0 (\mathrm{pres}(\mathcal{J}))$ in the Grothendieck group $K_0 (\mathrm{pres}(\mathcal{J}))$ of the abelian category $\mathrm{pres}(\mathcal{J})$. As an intermediate step we show that $\mathrm{pres}(\mathcal{J})$ is the abelianization of the (localized) category of complexes of $\mathbb{F}$-linear sheaves on $\mathbb{R}$, which are tame in the sense that sheaf cohomology of any open interval is finite-dimensional in each degree. This yields a close link between derived level set persistence by Curry, Kashiwara, and Schapira and the categorification of extended persistence diagrams.

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arXiv2022

A Unified View on the Functorial Nerve Theorem and its Variations

The nerve theorem is a basic result of algebraic topology that plays a central role in computational and applied aspects of the subject. In topological data analysis, one often needs a nerve theorem that is functorial in an appropriate sense, and furthermore one often needs a nerve theorem for closed covers as well as for open covers. While the techniques for proving such functorial nerve theorems have long been available, there is unfortunately no general-purpose, explicit treatment of this topic in the literature. We address this by proving a variety of functorial nerve theorems. First, we show how one can use elementary techniques to prove nerve theorems for covers by closed convex sets in Euclidean space, and for covers of a simplicial complex by subcomplexes. Then, we establish a more general, "unified" nerve theorem that subsumes many of the variants, using standard techniques from abstract homotopy theory.

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