arXiv subjects

Michael Kerber

Publications and source records attributed to Michael Kerber.

At least 19 recordsLinked to original sources

arXiv2026

Bifunction and Interlevel Delaunay Trifiltrations

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 $\mathbb{R}$-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 $\mathbb{R}^2$-valued function, also satisfying an analogous weak equivalence. For a point cloud $X \subset \mathbb{R}^d$, our trifiltration has size $O\bigl(|X|^{\lceil(d+1)/2\rceil+1}\bigr)$. We present an algorithm that computes this trifiltration in time $O\bigl(|X|^{\lceil d/2\rceil+2}\bigr)$, together with an implementation. Our experiments demonstrate that implementation can handle thousands of points in $\mathbb{R}^3$, with memory growth that is nearly linear.

cs.CG↗

arXiv2026

Topologically Stable Hough Transform

We propose an alternative formulation of the well-known Hough transform to detect lines in point clouds. Replacing the discretized voting scheme of the classical Hough transform by a continuous score function, its persistent features in the sense of persistent homology give a set of candidate lines. We also devise and implement an algorithm to efficiently compute these candidate lines.

cs.CG↗

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.

math.AT↗

arXiv2025

Computing the Bottleneck Distance between Persistent Homology Transforms

The Persistent Homology Transform (PHT) summarizes a shape in $\mathbb{R}^m$ by collecting persistence diagrams obtained from linear height filtrations in all directions on $\mathbb{S}^{m-1}$. It enjoys strong theoretical guarantees, including continuity, stability, and injectivity on broad classes of shapes. A natural way to compare two PHTs is to use the bottleneck distance between their diagrams as the direction varies. Prior work has either compared PHTs by sampling directions or, in 2D, computed the exact \textit{integral} of bottleneck distance over all angles via a kinetic data structure. We improve the integral objective to $\tilde O(n^5)$ in place of earlier $\tilde O(n^6)$ bound. For the \textit{max} objective, we give a $\tilde O(n^3)$ algorithm in $\mathbb{R}^2$ and a $\tilde O(n^5)$ algorithm in $\mathbb{R}^3$.

cs.CG↗

arXiv2025

Decomposing Multiparameter Persistence Modules

Dey and Xin (J.Appl.Comput.Top., 2022, arXiv:1904.03766) 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 \emph{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 with the same degree and with further optimisation we obtain an $O(n^3)$ 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 \textsc{aida} which is the first to enable the decomposition of large inputs. We show its capabilities via extensive experimental evaluation.

math.RT↗

arXiv2025

Representing two-parameter persistence modules via graphcodes

Graphcodes were recently introduced as a technique to employ two-parameter persistence modules in machine learning tasks (Kerber and Russold, NeurIPS 2024). We show in this work that a compressed version of graphcodes yields a description of a two-parameter module that is equivalent to a presentation of the module. This alternative representation as a graph allows for a simple translation between combinatorics and algebra: connected components of the graphcode correspond to summands of the module and isolated paths correspond to intervals. We demonstrate that graphcodes are useful in practice by speeding-up the task of decomposing a module into indecomposable summands. Also, the graphcode viewpoint allows to devise a simple algorithm to decide whether a persistence module is interval-decomposable in $O(n^4)$ time, which improves on the previous bound of $O(n^{2\omega+1})$.

math.AT↗

arXiv2024

Graphcode: Learning from multiparameter persistent homology using graph neural networks

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. We describe such a pipeline and demonstrate that graphcodes achieve better classification accuracy than state-of-the-art approaches on various datasets.

math.AT↗

arXiv2024

Experimental realisation of a universal inverse-design magnonic device

In the field of magnonics, which uses magnons, the quanta of spin waves, for energy-efficient data processing, significant progress has been made leveraging the capabilities of the inverse design concept. This approach involves defining a desired functionality and employing a feedback-loop algorithm to optimise the device design. In this study, we present the first experimental demonstration of a reconfigurable, lithography-free, and simulation-free inverse-design device capable of implementing various RF components. The device features a square array of independent direct current loops that generate a complex reconfigurable magnetic medium atop a Yttrium-Iron-Garnet (YIG) rectangular film for data processing in the gigahertz range. Showcasing its versatility, the device addresses inverse problems using two algorithms to create RF notch filters and demultiplexers. Additionally, the device holds promise for binary, reservoir, and neuromorphic computing applications.

physics.app-ph↗

arXiv2024

Probabilistic Analysis of Multiparameter Persistence Decompositions

Multiparameter persistence modules can be uniquely decomposed into indecomposable summands. Among these indecomposables, intervals stand out for their simplicity, making them preferable for their ease of interpretation in practical applications and their computational efficiency. Empirical observations indicate that modules that decompose into only intervals are rare. To support this observation, we show that for numerous common multiparameter constructions, such as density- or degree-Rips bifiltrations, and across a general category of point samples, the probability of the homology-induced persistence module decomposing into intervals goes to zero as the sample size goes to infinity.

math.AT↗

arXiv2023

Delaunay Bifiltrations of Functions on Point Clouds

The Delaunay filtration $\mathcal{D}_{\bullet}(X)$ of a point cloud $X\subset \mathbb{R}^d$ is a central tool of computational topology. Its use is justified by the topological equivalence of $\mathcal{D}_{\bullet}(X)$ and the offset (i.e., union-of-balls) filtration of $X$. Given a function $\gamma: X \to \mathbb{R}$, we introduce a Delaunay bifiltration $\mathcal{DC}_{\bullet}(\gamma)$ that satisfies an analogous topological equivalence, ensuring that $\mathcal{DC}_{\bullet}(\gamma)$ topologically encodes the offset filtrations of all sublevel sets of $\gamma$, as well as the topological relations between them. $\mathcal{DC}_{\bullet}(\gamma)$ is of size $O(|X|^{\lceil\frac{d+1}{2}\rceil})$, which for $d$ odd matches the worst-case size of $\mathcal{D}_{\bullet}(X)$. Adapting the Bowyer-Watson algorithm for computing Delaunay triangulations, we give a simple, practical algorithm to compute $\mathcal{DC}_{\bullet}(\gamma)$ in time $O(|X|^{\lceil \frac{d}{2}\rceil +1})$. Our implementation, based on CGAL, computes $\mathcal{DC}_{\bullet}(\gamma)$ with modest overhead compared to computing $\mathcal{D}_{\bullet}(X)$, and handles tens of thousands of points in $\mathbb{R}^3$ within seconds.

cs.CG↗

arXiv2023

Topological Data Analysis in smart manufacturing: State of the art and futuredirections

Topological Data Analysis (TDA) is a discipline that applies algebraic topology techniques to analyze complex, multi-dimensional data. Although it is a relatively new field, TDA has been widely and successfully applied across various domains, such as medicine, materials science, and biology. This survey provides an overview of the state of the art of TDA within a dynamic and promising application area: industrial manufacturing and production, particularly within the Industry 4.0 context. We have conducted a rigorous and reproducible literature search focusing on TDA applications in industrial production and manufacturing settings. The identified works are categorized based on their application areas within the manufacturing process and the types of input data. We highlight the principal advantages of TDA tools in this context, address the challenges encountered and the future potential of the field. Furthermore, we identify TDA methods that are currently underexploited in specific industrial areas and discuss how their application could be beneficial, with the aim of stimulating further research in this field. This work seeks to bridge the theoretical advancements in TDA with the practical needs of industrial production. Our goal is to serve as a guide for practitioners and researchers applying TDA in industrial production and manufacturing systems. We advocate for the untapped potential of TDA in this domain and encourage continued exploration and research.

cs.LG↗

arXiv2023

The Localized Union-of-Balls Bifiltration

We propose an extension of the classical union-of-balls filtration of persistent homology: fixing a point $q$, we focus our attention to a ball centered at $q$ whose radius is controlled by a second scale parameter. We discuss an absolute variant, where the union is just restricted to the $q$-ball, and a relative variant where the homology of the $q$-ball relative to its boundary is considered. Interestingly, these natural constructions lead to bifiltered simplicial complexes which are not $k$-critical for any finite $k$. Nevertheless, we demonstrate that these bifiltrations can be computed exactly and efficiently, and we provide a prototypical implementation using the CGAL library. We also argue that some of the recent algorithmic advances for $2$-parameter persistence (which usually assume $k$-criticality for some finite $k$) carry over to the $\infty$-critical case.

cs.CG↗

arXiv2023

Sparse Higher Order \v{C}ech Filtrations

For a finite set of balls of radius $r$, the $k$-fold cover is the space covered by at least $k$ balls. Fixing the ball centers and varying the radius, we obtain a nested sequence of spaces that is called the $k$-fold filtration of the centers. For $k=1$, the construction is the union-of-balls filtration that is popular in topological data analysis. For larger $k$, it yields a cleaner shape reconstruction in the presence of outliers. We contribute a sparsification algorithm to approximate the topology of the $k$-fold filtration. Our method is a combination and adaptation of several techniques from the well-studied case $k=1$, resulting in a sparsification of linear size that can be computed in expected near-linear time with respect to the number of input points. Our method also extends to the multicover bifiltration, composed of the $k$-fold filtrations for several values of $k$, with the same size and complexity bounds.

cs.CG↗

arXiv2023

Decomposition of zero-dimensional persistence modules via rooted subsets

We study the decomposition of zero-dimensional persistence modules, viewed as functors valued in the category of vector spaces factorizing through sets. Instead of working directly at the level of vector spaces, we take a step back and first study the decomposition problem at the level of sets. This approach allows us to define the combinatorial notion of rooted subsets. In the case of a filtered metric space $M$, rooted subsets relate the clustering behavior of the points of $M$ with the decomposition of the associated persistence module. In particular, we can identify intervals in such a decomposition quickly. In addition, rooted subsets can be understood as a generalization of the elder rule, and are also related to the notion of constant conqueror of Cai, Kim, M\'emoli and Wang. As an application, we give a lower bound on the number of intervals that we can expect in the decomposition of zero-dimensional persistence modules of a density-Rips filtration in Euclidean space: in the limit, and under very general circumstances, we can expect that at least 25% of the indecomposable summands are interval modules.

math.AT↗

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

Filtration-Domination in Bifiltered Graphs

Bifiltered graphs are a versatile tool for modelling relations between data points across multiple grades of a two-dimensional scale. They are especially popular in topological data analysis, where the homological properties of the induced clique complexes are studied. To reduce the large size of these clique complexes, we identify filtration-dominated edges of the graph, whose removal preserves the relevant topological properties. We give two algorithms to detect filtration-dominated edges in a bifiltered graph and analyze their complexity. These two algorithms work directly on the bifiltered graph, without first extracting the clique complexes, which are generally much bigger. We present extensive experimental evaluation which shows that in most cases, more than 90% of the edges can be removed. In turn, we demonstrate that this often leads to a substantial speedup, and reduction in the memory usage, of the computational pipeline of multiparameter topological data analysis.

cs.CG↗

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.

math.AT↗

arXiv2021

Asymptotic Improvements on the Exact Matching Distance for 2-parameter Persistence

In the field of topological data analysis, persistence modules are used to express geometrical features of data sets. The matching distance $d_\mathcal{M}$ measures the difference between $2$-parameter persistence modules by taking the maximum bottleneck distance between $1$-parameter slices of the modules. The previous best algorithm to compute $d_\mathcal{M}$ exactly runs in $O(n^{8+\omega})$ time using $O(n^4)$ space, where $n$ is the number of generators and relations of the modules and $\omega$ is the matrix multiplication constant. We improve significantly on this by describing an algorithm with expected running time $O(n^5 \log^3 n)$ and using $O(n^2)$ space. We first solve the decision problem $d_\mathcal{M}\leq \lambda$ for a constant $\lambda$ in $O(n^5\log n)$ time by traversing a line arrangement in the dual plane, where each point represents a slice. Then we lift the line arrangement to a plane arrangement in $\mathbb{R}^3$ whose vertices represent possible values for $d_\mathcal{M}$, and use a randomized incremental method to search through the vertices and find $d_\mathcal{M}$. The expected running time of this algorithm is $O((n^4+T(n))\log^2 n)$, where $T(n)$ is an upper bound for the complexity of deciding if $d_\mathcal{M}\leq \lambda$. Moreover, we show how to compute the matching distance using only linear space, to the price of a much worse time complexity.

math.AT↗