arXiv subjects

Florian Russold

Publications and source records attributed to Florian Russold.

9 recordsLinked to original sources

arXiv2026

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

Computing Projective Implicit Representations from Poset Towers

A family of simplicial complexes connected by simplicial maps and indexed by a finite poset $P$ is called a poset tower. Poset towers subsume multi-parameter filtrations, zigzag filtrations, and one-parameter simplicial towers, while allowing arbitrary finite posets and simplicial maps. The homology of a poset tower is a $P$-persistence module. To compute it globally over $P$, we consider the chain complex segment of $P$-persistence modules $C_{\ell-1}\xleftarrow{\partial_{\ell}}C_\ell \xleftarrow{\partial_{\ell+1}}C_{\ell+1}$ induced by the simplices of the tower. Unlike in one-critical multi-filtrations, the chain modules $C_\ell$ need not be projective and may have a complicated structure. We address the problem of replacing this segment by projective modules and $P$-graded matrices while preserving homology. The resulting projective implicit representation (PiRep) plays the role of the graded boundary-matrix representation in the classical persistence algorithm: it converts simplicial data into algebraic input on which persistent homology can be computed globally over $P$. In particular, a PiRep can be used as input to algorithms for computing minimal presentations of persistent homology. We give an efficient algorithm to compute a PiRep from a poset tower. It constructs degreewise minimal presentations and asymptotically minimal second terms of projective resolutions of the chain modules $C_\ell$, lifts the boundary maps $\partial_\ell$ to these resolutions, and assembles the resulting data into a PiRep using an additional correction term. The method is tailored to chain complexes induced by poset towers and computes the required algebraic data combinatorially, exploiting their special structure and avoiding general-purpose algebraic reduction. In the context of poset towers, it is fully general and can serve as a foundation for efficient algorithms on specific posets.

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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 $O(n^{2ω+1})$.

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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.

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arXiv2024

Stability and Approximations for Decorated Reeb Spaces

Given a map $f:X \to M$ from a topological space $X$ to a metric space $M$, a decorated Reeb space consists of the Reeb space, together with an attribution function whose values recover geometric information lost during the construction of the Reeb space. For example, when $M=\mathbb{R}$ is the real line, the Reeb space is the well-known Reeb graph, and the attributions may consist of persistence diagrams summarizing the level set topology of $f$. In this paper, we introduce decorated Reeb spaces in various flavors and prove that our constructions are Gromov-Hausdorff stable. We also provide results on approximating decorated Reeb spaces from finite samples and leverage these to develop a computational framework for applying these constructions to point cloud data.

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arXiv2024

Efficient Algorithms for Complexes of Persistence Modules with Applications

We extend the persistence algorithm, viewed as an algorithm computing the homology of a complex of free persistence or graded modules, to complexes of modules that are not free. We replace persistence modules by their presentations and develop an efficient algorithm to compute the homology of a complex of presentations. To deal with inputs that are not given in terms of presentations, we give an efficient algorithm to compute a presentation of a morphism of persistence modules. This allows us to compute persistent (co)homology of instances giving rise to complexes of non-free modules. Our methods lead to a new efficient algorithm for computing the persistent homology of simplicial towers and they enable efficient algorithms to compute the persistent homology of cosheaves over simplicial towers and cohomology of persistent sheaves on simplicial complexes. We also show that we can compute the cohomology of persistent sheaves over arbitrary finite posets by reducing the computation to a computation over simplicial complexes.

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arXiv2023

Topologically Attributed Graphs for Shape Discrimination

In this paper we introduce a novel family of attributed graphs for the purpose of shape discrimination. Our graphs typically arise from variations on the Mapper graph construction, which is an approximation of the Reeb graph for point cloud data. Our attributions enrich these constructions with (persistent) homology in ways that are provably stable, thereby recording extra topological information that is typically lost in these graph constructions. We provide experiments which illustrate the use of these invariants for shape representation and classification. In particular, we obtain competitive shape classification results when using our topologically attributed graphs as inputs to a simple graph neural network classifier.

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arXiv2023

Convergence of Leray Cosheaves for Decorated Mapper Graphs

We introduce decorated mapper graphs as a generalization of mapper graphs capable of capturing more topological information of a data set. A decorated mapper graph can be viewed as a discrete approximation of the cellular Leray cosheaf over the Reeb graph. We establish a theoretical foundation for this construction by showing that the cellular Leray cosheaf with respect to a sequence of covers converges to the actual Leray cosheaf as the resolution of the covers goes to zero.

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arXiv2022

Persistent sheaf cohomology

We expand the toolbox of (co)homological methods in computational topology by applying the concept of persistence to sheaf cohomology. Since sheaves (of modules) combine topological information with algebraic information, they allow for variation along an algebraic dimension and along a topological dimension. Consequently, we introduce two different constructions of sheaf cohomology (co)persistence modules. One of them can be viewed as a natural generalization of the construction of simplicial or singular cohomology copersistence modules. We discuss how both constructions relate to each other and show that, in some cases, we can reduce one of them to the other. Moreover, we show that we can combine both constructions to obtain two-dimensional (co)persistence modules with a topological and an algebraic dimension. We also show that some classical results and methods from persistence theory can be generalized to sheaves. Our results open up a new perspective on persistent cohomology of filtrations of simplicial complexes.

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