arXiv subjects

Magnus Bakke Botnan

Publications and source records attributed to Magnus Bakke Botnan.

At least 19 recordsLinked to original sources

arXiv2026

Cheeger Inequalities for the Persistent Laplacian

We study Cheeger-type inequalities for persistent Laplacians associated with inclusions of simplicial complexes $\mathcal{K}\hookrightarrow \mathcal{L}$. We introduce a persistent up $p$-Laplacian $Δ_{q,p,\mathrm{up}}^{\mathcal{K},\mathcal{L}}$ for $p\geq 1$. For $p=2$, this recovers the usual persistent up Laplacian, while for $p=1$ it yields a nonzero persistent Cheeger constant $φ_q^{\mathcal{K},\mathcal{L}}$. We prove a Cheeger-type inequality relating $φ_q^{\mathcal{K},\mathcal{L}}$ to the smallest nonzero eigenvalue of $Δ_{q,\mathrm{up}}^{\mathcal{K},\mathcal{L}}$. This gives a persistent extension of recent work by Jost and Zhang (arXiv:2302.01069). We then study two more structured settings. Under a locally complete $q$-skeleton assumption on $\mathcal{K}$, we extend the complete-skeleton isoperimetric inequality of Parzanchevski--Rosenthal--Tessler (arXiv:1207.0638) to the persistent setting. For orientable $(q+1)$-dimensional pseudomanifolds, we prove a Kron-type reduction of the persistent up Laplacian to a vertex- and edge-weighted graph Laplacian, possibly with Dirichlet boundary terms, and obtain two-sided Cheeger inequalities; this is related to the dual-graph perspective in the work of Steenbergen--Klivans--Mukherjee (arXiv:1209.5091). We also describe the nonzero persistent Cheeger constant $φ_q^{\mathcal{K},\mathcal{L}}$ explicitly in terms of the dual graph in the non-branching pseudomanifold case. Finally, we specialize our two constructions to graph inclusions and compare them with the graph-pair theory of Mémoli--Wan--Wang (arXiv:2012.02808). We establish precise relationships between the two persistent Cheeger constants arising from our constructions and the corresponding constants derived via Kron reduction.

math.AT↗

arXiv2026

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.

math.RT↗

arXiv2025

The persistent Laplacian of non-branching complexes

Non-branching matrices are real matrices with entries in $\{-1,0,1\}$, where each row contains at most two non-zero entries. Such matrices naturally arise in the study of Laplacians of pseudomanifolds and cubical complexes. We show that a basis for the kernel of a non-branching matrix can be computed in near-linear time. Especially, the basis has the special property that the supports of all the column vectors in it are disjoint with each other. Building on this result, we show that the up persistent Laplacian can be computed in near-linear time for a pair of such spaces and its eigenvalues can be more efficiently computed via computing singular values. In addition to that, we analyze the arithmetic operations of up persistent Laplacian with respect to a non-branching filtration. Furthermore, we show that the up persistent Laplacian of $q$-non-branching simplicial complexes can be represented as the Laplacian of an associated hypergraph, thus providing a higher-dimensional generalization of the Kron reduction, as well as a Cheeger-type inequality. Finally, we highlight the efficiency of our method on image data.

math.AT↗

arXiv2025

Computing $p$-presentation distances is hard

Recently, $p$-presentation distances for $p\in [1,\infty]$ were introduced for merge trees and multiparameter persistence modules as more sensitive variations of the respective interleaving distances ($p=\infty)$. It is well-known that computing the interleaving distance is NP-hard in both cases. We extend this result by showing that computing the $p$-presentation distance is NP-hard for all $p\in [1,\infty)$ for both merge trees and $t$-parameter persistence modules for any $t\geq 2$. Though the details differ, both proofs follow the same novel strategy, suggesting that our approach can be adapted to proving the NP-hardness of other distances based on sums or $p$-norms.

cs.CG↗

arXiv2025

Extremal Betti Numbers and Persistence in Flag Complexes

We investigate several problems concerning extremal Betti numbers and persistence in filtrations of flag complexes. For graphs on $n$ vertices, we show that $β_k(X(G))$ is maximal when $G=\mathcal{T}_{n,k+1}$, the Turán graph on $k+1$ partition classes, where $X(G)$ denotes the flag complex of $G$. Building on this, we construct an edgewise (one edge at a time) filtration $\mathcal{G}=G_1\subseteq \cdots \subseteq \mathcal{T}_{n,k+1}$ for which $β_k(X(G_i))$ is maximal for all graphs on $n$ vertices and $i$ edges. Moreover, the persistence barcode $\mathcal{B}_k(X(G))$ achieves a maximal number of intervals, and total persistence, among all edgewise filtrations with $|E(\mathcal{T}_{n,k+1})|$ edges. For $k=1$, we consider edgewise filtrations of the complete graph $K_n$. We show that the maximal number of intervals in the persistence barcode is obtained precisely when $G_{\lceil n/2\rceil \cdot \lfloor n/2 \rfloor}=\mathcal{T}_{n,2}$. Among such filtrations, we characterize those achieving maximal total persistence. We further show that no filtration can optimize $β_1(X(G_i))$ for all $i$, and conjecture that our filtrations maximize the total persistence over all edgewise filtrations of $K_n$.

math.CO↗

arXiv2024

On the bottleneck stability of rank decompositions of multi-parameter persistence modules

A significant part of modern topological data analysis is concerned with the design and study of algebraic invariants of poset representations -- often referred to as multi-parameter persistence modules. One such invariant is the minimal rank decomposition, which encodes the ranks of all the structure morphisms of the persistence module by a single ordered pair of rectangle-decomposable modules, interpreted as a signed barcode. This signed barcode generalizes the concept of persistence barcode from one-parameter persistence to any number of parameters, raising the question of its bottleneck stability. We show in this paper that the minimal rank decomposition is not stable under the natural notion of signed bottleneck matching between signed barcodes. We remedy this by turning our focus to the rank exact decomposition, a related signed barcode induced by the minimal projective resolution of the module relative to the so-called rank exact structure, which we prove to be bottleneck stable under signed matchings. As part of our proof, we obtain two intermediate results of independent interest: we compute the global dimension of the rank exact structure on the category of finitely presentable multi-parameter persistence modules, and we prove a bottleneck stability result for hook-decomposable modules. We also give a bound for the size of the rank exact decomposition that is polynomial in the size of the usual minimal projective resolution, we prove a universality result for the dissimilarity function induced by the notion of signed matching, and we compute, in the two-parameter case, the global dimension of a different exact structure related to the upsets of the indexing poset. This set of results combines concepts from topological data analysis and from the representation theory of posets, and we believe is relevant to both areas.

math.AT↗

arXiv2024

Signed Barcodes for Multi-Parameter Persistence via Rank Decompositions and Rank-Exact Resolutions

In this paper, we introduce the signed barcode, a new visual representation of the global structure of the rank invariant of a multi-parameter persistence module or, more generally, of a poset representation. Like its unsigned counterpart in one-parameter persistence, the signed barcode decomposes the rank invariant as a $\Z$-linear combination of rank invariants of indicator modules supported on segments in the poset. We develop the theory behind these decompositions, both for the usual rank invariant and for its generalizations, showing under what conditions they exist and are unique. We also show that, like its unsigned counterpart, the signed barcode reflects in part the algebraic structure of the module: specifically, it derives from the terms in the minimal rank-exact resolution of the module, i.e., its minimal projective resolution relative to the class of short exact sequences on which the rank invariant is additive. To complete the picture, we show some experimental results that illustrate the contribution of the signed barcode in the exploration of multi-parameter persistence modules.

math.AT↗

arXiv2024

Universal Distances for Extended Persistence

The extended persistence diagram is an invariant of piecewise linear functions, which is known to be stable under perturbations of functions with respect to the bottleneck distance as introduced by Cohen-Steiner, Edelsbrunner, and Harer. We address the question of universality, which asks for the largest possible stable distance on extended persistence diagrams, showing that a more discriminative variant of the bottleneck distance is universal. Our result applies more generally to settings where persistence diagrams are considered only up to a certain degree. We achieve our results by establishing a functorial construction and several characteristic properties of relative interlevel set homology, which mirror the classical Eilenberg--Steenrod axioms. Finally, we contrast the bottleneck distance with the interleaving distance of sheaves on the real line by showing that the latter is not intrinsic, let alone universal. This particular result has the further implication that the interleaving distance of Reeb graphs is not intrinsic either.

math.AT↗

arXiv2024

Stable Vectorization of Multiparameter Persistent Homology using Signed Barcodes as Measures

Persistent homology (PH) provides topological descriptors for geometric data, such as weighted graphs, which are interpretable, stable to perturbations, and invariant under, e.g., relabeling. Most applications of PH focus on the one-parameter case -- where the descriptors summarize the changes in topology of data as it is filtered by a single quantity of interest -- and there is now a wide array of methods enabling the use of one-parameter PH descriptors in data science, which rely on the stable vectorization of these descriptors as elements of a Hilbert space. Although the multiparameter PH (MPH) of data that is filtered by several quantities of interest encodes much richer information than its one-parameter counterpart, the scarceness of stability results for MPH descriptors has so far limited the available options for the stable vectorization of MPH. In this paper, we aim to bring together the best of both worlds by showing how the interpretation of signed barcodes -- a recent family of MPH descriptors -- as signed measures leads to natural extensions of vectorization strategies from one parameter to multiple parameters. The resulting feature vectors are easy to define and to compute, and provably stable. While, as a proof of concept, we focus on simple choices of signed barcodes and vectorizations, we already see notable performance improvements when comparing our feature vectors to state-of-the-art topology-based methods on various types of data.

cs.LG↗

arXiv2023

An Introduction to Multiparameter Persistence

In topological data analysis (TDA), one often studies the shape of data by constructing a filtered topological space, whose structure is then examined using persistent homology. However, a single filtered space often does not adequately capture the structure of interest in the data, and one is led to consider multiparameter persistence, which associates to the data a space equipped with a multiparameter filtration. Multiparameter persistence has become one of the most active areas of research within TDA, with exciting progress on several fronts. In this article, we introduce multiparameter persistence and survey some of this recent progress, with a focus on ideas likely to lead to practical applications in the near future.

math.AT↗

arXiv2022

Local characterizations for decomposability of 2-parameter persistence modules

We investigate the existence of sufficient local conditions under which poset representations decompose as direct sums of indecomposables from a given class. In our work, the indexing poset is the product of two totally ordered sets, corresponding to the setting of 2-parameter persistence in topological data analysis. Our indecomposables of interest belong to the so-called interval modules, which by definition are indicator representations of intervals in the poset. While the whole class of interval modules does not admit such a local characterization, we show that the subclass of rectangle modules does admit one and that it is, in some precise sense, the largest subclass to do so.

math.RT↗

arXiv2022

Structure and Interleavings of Relative Interlevel Set Cohomology

The relative interlevel set cohomology (RISC) is an invariant of real-valued continuous functions closely related to the Mayer--Vietoris pyramid introduced by Carlsson, de Silva, and Morozov. As such, the relative interlevel set cohomology is a parametrization of the cohomology vector spaces of all open interlevel sets relative complements of closed interlevel sets. We provide a structure theorem, which applies to the RISC of real-valued continuous functions whose open interlevel sets have finite-dimensional cohomology in each degree. Moreover, we show this tameness assumption is in some sense equivalent to $q$-tameness as introduced by Chazal, de Silva, Glisse, and Oudot. Furthermore, we provide the notion of an interleaving for RISC and we show that it is stable in the sense that any space with two functions that are $δ$-close induces a $δ$-interleaving of the corresponding relative interlevel set cohomologies. Finally, we provide an elementary form of quantitative homotopy invariance for RISC.

math.AT↗

arXiv2022

On rectangle-decomposable 2-parameter persistence modules

This paper addresses two questions: (a) can we identify a sensible class of 2-parameter persistence modules on which the rank invariant is complete? (b) can we determine efficiently whether a given 2-parameter persistence module belongs to this class? We provide positive answers to both questions, and our class of interest is that of rectangle-decomposable modules. Our contributions include: on the one hand, a proof that the rank invariant is complete on rectangle-decomposable modules, together with an inclusion-exclusion formula for counting the multiplicities of the summands; on the other hand, algorithms to check whether a module induced in homology by a bifiltration is rectangle-decomposable, and to decompose it in the affirmative, with a better complexity than state-of-the-art decomposition methods for general 2-parameter persistence modules. Our algorithms are backed up by a new structure theorem, whereby a 2-parameter persistence module is rectangle-decomposable if, and only if, its restrictions to squares are. This local characterization is key to the efficiency of our algorithms, and it generalizes previous conditions derived for the smaller class of block-decomposable modules. It also admits an algebraic formulation that turns out to be a weaker version of the one for block-decomposability. By contrast, we show that general interval-decomposability does not admit such a local characterization, even when locality is understood in a broad sense. Our analysis focuses on the case of modules indexed over finite grids, the more general cases are left as future work.

math.AT↗

arXiv2021

On the consistency and asymptotic normality of multiparameter persistent Betti numbers

The persistent Betti numbers are used in topological data analysis to infer the scales at which topological features appear and disappear in the filtration of a topological space. Most commonly by means of the corresponding barcode or persistence diagram. While this approach to data science has been very successful, it suffers from sensitivity to outliers, and it does not allow for additional filtration parameters. Such parameters naturally appear when a cloud of data points comes together with additional measurements taken at the locations of the data. For these reasons, multiparameter persistent homology has recently received significant attention. In particular, the multicover and Čech bifiltration have been introduced to overcome the aforementioned shortcomings. In this work, we establish the strong consistency and asymptotic normality of the multiparameter persistent Betti numbers in growing domains. Our asymptotic results are established for a general framework encompassing both the marked Čech bifiltration, as well as the multicover bifiltration constructed on the null model of an independently marked Poisson point process. In a simulation study, we explain how the asymptotic normality can be used to derive tests for the goodness of fit. The statistical power of such tests is illustrated through different alternatives exhibiting more clustering, or more repulsion than the null model.

math.ST↗

arXiv2020

A Relative Theory of Interleavings

The interleaving distance, although originally developed for persistent homology, has been generalized to measure the distance between functors modeled on many posets or even small categories. Existing theories require that such a poset have a superlinear family of translations or a similar structure. However, many posets of interest to topological data analysis, such as zig-zag posets and the face relation poset of a cell-complex, do not admit interesting translations, and consequently don't admit a nice theory of interleavings. In this paper we show how one can side-step this limitation by providing a general theory where one maps to a poset that does admit interesting translations, such as the lattice of down sets, and then defines interleavings relative to this map. Part of our theory includes a rigorous notion of discretization or "pixelization" of poset modules, which in turn we use for interleaving inference. We provide an approximation condition that in the setting of lattices gives rise to two possible pixelizations, both of which are guaranteed to be close in the interleaving distance. Finally, we conclude by considering interleaving inference for cosheaves over a metric space and give an explicit description of interleavings over a grid structure on Euclidean space.

math.CT↗

arXiv2019

Computing the interleaving distance is NP-hard

We show that computing the interleaving distance between two multi-graded persistence modules is NP-hard. More precisely, we show that deciding whether two modules are $1$-interleaved is NP-complete, already for bigraded, interval decomposable modules. Our proof is based on previous work showing that a constrained matrix invertibility problem can be reduced to the interleaving distance computation of a special type of persistence modules. We show that this matrix invertibility problem is NP-complete. We also give a slight improvement of the above reduction, showing that also the approximation of the interleaving distance is NP-hard for any approximation factor smaller than $3$. Additionally, we obtain corresponding hardness results for the case that the modules are indecomposable, and in the setting of one-sided stability. Furthermore, we show that checking for injections (resp. surjections) between persistence modules is NP-hard. In conjunction with earlier results from computational algebra this gives a complete characterization of the computational complexity of one-sided stability. Lastly, we show that it is in general NP-hard to approximate distances induced by noise systems within a factor of 2.

cs.CG↗

arXiv2019

Decomposition of persistence modules

We show that a pointwise finite-dimensional persistence module indexed over a small category decomposes into a direct sum of indecomposables with local endomorphism rings. As an application of this result we give new, short proofs of fundamental structure theorems for persistence modules.

math.RT↗

arXiv2018

Computational Complexity of the Interleaving Distance

The interleaving distance is arguably the most prominent distance measure in topological data analysis. In this paper, we provide bounds on the computational complexity of determining the interleaving distance in several settings. We show that the interleaving distance is NP-hard to compute for persistence modules valued in the category of vector spaces. In the specific setting of multidimensional persistent homology we show that the problem is at least as hard as a matrix invertibility problem. Furthermore, this allows us to conclude that the interleaving distance of interval decomposable modules depends on the characteristic of the field. Persistence modules valued in the category of sets are also studied. As a corollary, we obtain that the isomorphism problem for Reeb graphs is graph isomorphism complete.

cs.CG↗