arXiv subjects

Morten Brun

Publications and source records attributed to Morten Brun.

At least 19 recordsLinked to original sources

arXiv2025

Monoidal Rips: Stable Multiparameter Filtrations of Directed Networks

We introduce the monoidal Rips filtration, a filtered simplicial set for weighted directed graphs and other lattice-valued networks. Our construction generalizes the Vietoris-Rips filtration for metric spaces by replacing the maximum operator, determining the filtration values, with a more general monoidal product. We establish interleaving guarantees for the monoidal Rips persistent homology, capturing existing stability results for real-valued networks. When the lattice is a product of totally ordered sets, we are in the setting of multiparameter persistence. Here, the interleaving distance is bounded in terms of a generalized network distance. We use this to prove a novel stability result for the sublevel Rips bifiltration. Our experimental results show that our method performs better than Flagser in a graph regression task, and that combining different monoidal products in point cloud classification can improve performance.

math.AT↗

arXiv2025

Flexible inference of evolutionary accumulation dynamics using uncertain observational data

Understanding and predicting evolutionary accumulation pathways is a key objective in many fields of research, ranging from classical evolutionary biology to diverse applications in medicine. In this context, we are often confronted with the problem that data is sparse and uncertain. To use the available data as best as possible, inference approaches that can handle this uncertainty are required. One way that allows us to use not only cross-sectional data, but also phylogenetic related and longitudinal data, is using `hypercubic inference' models. In this article we introduce HyperLAU, a new algorithm for hypercubic inference that makes it possible to use datasets including uncertainties for learning evolutionary pathways. Expanding the flexibility of accumulation modelling, HyperLAU allows us to infer dynamic pathways and interactions between features, even when large sets of particular features are unobserved across the source dataset. We show that HyperLAU is able to highlight the main pathways found by other tools, even when up to 50% of the features in the input data are uncertain. Additionally, we demonstrate how it can help to overcome possible biases that can occur then reducing the used data by excluding uncertain parts. We illustrate the approach with a case study on multidrug resistance in tuberculosis, showing that HyperLAU allows more flexible data and provides new information about evolutionary pathways compared to existing approaches.

q-bio.PE↗

arXiv2024

Random Connection Hypergraphs

In this paper, we introduce a novel model for random hypergraphs based on weighted random connection models. In accordance with the standard theory for hypergraphs, this model is constructed from a bipartite graph. In our stochastic model, both vertex sets of this bipartite graph form marked Poisson point processes, and the connection radius is inversely proportional to a product of suitable powers of the marks. Hence, our model is a common generalization of weighted random connection models and AB random geometric graphs. For this hypergraph model, we investigate the limit theory of various graph-theoretic and topological characteristics, including higher-order degree distributions, Betti numbers of the associated Dowker complex, and simplex counts. In particular, for the latter quantity, we identify regimes of convergence to normal and to stable distribution depending on the heavy-tailedness of the weight distribution. We conclude our investigation with a simulation study and an application to the collaboration network extracted from the arXiv dataset.

math.PR↗

arXiv2024

The Dowker theorem via discrete Morse theory

The Dowker theorem is a classical result in the topology of finite spaces, claiming that any binary relation between two finite spaces defines two homotopy-equivalent complexes (the Dowker complexes). Recently, Barmak strengthened this to a simple-homotopy-equivalence. We reprove Barmak's result using a combinatorial argument that constructs an explicit acyclic matching in the sense of discrete Morse theory.

math.CO↗

arXiv2024

The Dual Degree Cech Bifiltration

In topological data analysis (TDA), a longstanding challenge is to recognize underlying geometric structures in noisy data. One motivating examples is the shape of a point cloud in Euclidean space given by image. Carlsson et al. proposed a method to detect topological features in point clouds by first filtering by density and then applying persistent homology. Later more refined methods have been developed, such as the degree Rips complex of Lesnick and Wright and the multicover bifiltration. In this paper we introduce the dual Degree Cech bifiltration, a Prohorov stable bicomplex of a point cloud in a metric space with the point cloud itself as vertex set. It is of the same homotopy type as the Measure Dowker bifiltration of Hellmer and Spali\'nski but it has a different vertex set. The dual Degree Cech bifiltration can be constructed both in an ambient and an intrinsic way. The intrinsic dual Degree Cech bifiltration is a $(1,2)$-intereaved with the ambent dual Degree Cech bifiltration in the distance parameter. This interleaving can be used to leverage a stability result for the intrinsically defined dual Degree Cech bifiltration. This stability result recently occured in work by Hellmer and Spali\'nski.

math.AT↗

arXiv2024

Core Bifiltration

The motivation of this paper is to recognize a geometric shape from a noisy sample in the form of a point cloud. Inspired by the HDBSCAN clustering algorithm, we introduce the core dissimilarity, from which we construct the core bifiltration. We also consider the Delaunay core bifiltration by intersecting with Voronoi cells, giving us a filtered simplicial complex of smaller size. A major advantage of the (Delaunay) core bifiltration is that, for each filtration value, it admits a good cover of balls. By the persistent nerve theorem, the nerve of this cover is homotopy equivalent to the (Delaunay) core bifiltration. We show that the multicover-, core- and Delaunay core bifiltrations are all interleaved, and that they enjoy similar stability properties with respect to the Prohorov distance. We have performed experiments with the Delaunay core bifiltration. In the experiments, we calculated persistent homology along lines in the two-dimensional persistence parameter space, as well as multipersistence module approximations and Hilbert functions for the full Delaunay core bifiltration.

math.AT↗

arXiv2023

Dowker Duality for Relations of Categories

We propose a categorification of the Dowker duality theorem for relations. Dowker's theorem states that the Dowker complex of a relation $R \subseteq X \times Y$ of sets $X$ and $Y$ is homotopy equivalent to the Dowker complex of the transpose relation $R^T \subseteq Y \times X$. Given a relation $R$ of small categories $\mathcal{C}$ and $\mathcal{D}$, that is, a functor of the form $R \colon \mathcal{R} \to \mathcal{C} \times \mathcal{D}$, we define the bisimplicial rectangle nerve $ER$ and the Dowker nerve $DR$. The diagonal $d(ER)$ of the bisimplicial set $ER$ maps to the simplicial set $DR$ by a natural projection $d(\pi_R) \colon d(ER) \to DR$. We introduce a criterion on relations of categories ensuring that the projection from the diagonal of the bisimplicial rectangle nerve to the Dowker nerve is a weak equivalence. Relations satisfying this criterion are called Dowker relations. If both the relation $R$ of categories and its transpose relation $R^T$ are Dowker relations, then the Dowker nerves $DR$ and $DR^T$ are weakly equivalent simplicial sets. In order to justify the abstraction introduced by our categorification we give two applications. The first application is to show that Quillen's Theorem A can be considered as an instance of Dowker duality. In the second application we consider a simplicial complex $K$ with vertex set $V$ and show that the geometric realization of $K$ is naturally homotopy equivalent to the geometric realization of the simplicial set with the set of $n$-simplices given by functions $\{0,1,\dots,n\}\to V$ whose image is a simplex of $K$.

math.AT↗

arXiv2022

The Rectangle Complex of a Relation

We construct a simplicial complex, the rectangle complex of a relation R, and show that it is homotopy equivalent to the Dowker complex of R. This results in a short and conceptual proof of functorial versions of Dowker's Theorem used in topological data analysis.

math.AT↗

arXiv2022

Equivariant Structure on Smash Powers

We provide foundations for dealing with the equivariant structure of "smash powers" of commutative orthogonal ring spectra. The category of commutative orthogonal ring spectra $A$ is tensored over spaces $X$, so that $A \otimes X$ is a commutative orthogonal ring spectrum. If $X$ is a discrete space, this is literally the smash power of $A$ with itself indexed over $X$, and we keep this language also in the nondiscrete case. In particular $A \otimes S^1$ is a model for topological Hochschild homology. We provide a framework where a generalization of the cyclotomic structure of topological Hochschild homology is visible in a categorical framework, also for more general $G$ and $X$. Similar situations have been studied by others, e.g., in Hill, Hopkins and Ravenel's treatment of the norm construction and Brun, Carlsson, Dundas' covering homology. In the case of non-commutative $A$ and $X=S^1$, the situation is somewhat easier and has already been covered by Kro. We are motivated by applications to $G$ being a torus in order to study the iterated algebraic $K$-theory, and have to develop a categorical theory that in some ways goes beyond what has been done before. Most of the material appeared in the last author's thesis which was defended in 2011. We apologize for the delay.

math.AT↗

arXiv2021

Determining homology of an unknown space from a sample

The homology of an unknown subspace of Euclidean space can be determined from the intrinsic Čech complex of a sample of points in the subspace, without reference to the ambient Euclidean space. More precisely, given a subspace $X$ of Euclidean space and a sample $A$ of points in $X$, we give conditions for the homology of $X$ to be isomorphic to a certain persistent homology group of the intrinsic Čech complex.

math.AT↗

arXiv2021

The Parameterized Complexity of Finding Minimum Bounded Chains

Finding the smallest $d$-chain with a specific $(d-1)$-boundary in a simplicial complex is known as the \textsc{Minimum Bounded Chain} (MBC$_d$) problem. The MBC$_d$ problem is NP-hard for all $d\geq 2$. In this paper, we prove that it is also W[1]-hard for all $d\geq 2$, if we parameterize the problem by solution size. We also give an algorithm solving the MBC$_1$ problem in polynomial time and introduce and implemented two fixed parameter tractable (FPT) algorithms solving the MBC$_d$ problem for all $d$. The first algorithm is a generalized version of Dijkstra's algorithm and is parameterized by solution size and coface degree. The second algorithm is a dynamic programming approach based on treewidth, which has the same runtime as a lower bound we prove under the exponential time hypothesis.

cs.CG↗

arXiv2021

A Comparison of the Delta Method and the Bootstrap in Deep Learning Classification

We validate the recently introduced deep learning classification adapted Delta method by a comparison with the classical Bootstrap. We show that there is a strong linear relationship between the quantified predictive epistemic uncertainty levels obtained from the two methods when applied on two LeNet-based neural network classifiers using the MNIST and CIFAR-10 datasets. Furthermore, we demonstrate that the Delta method offers a five times computation time reduction compared to the Bootstrap.

cs.LG↗

arXiv2021

Efficient Computation of Hessian Matrices in TensorFlow

The Hessian matrix has a number of important applications in a variety of different fields, such as optimzation, image processing and statistics. In this paper we focus on the practical aspects of efficiently computing Hessian matrices in the context of deep learning using the Python scripting language and the TensorFlow library. We define a general feed-forward neural network model and show how to efficiently compute two quantities: the cost function's exact Hessian matrix, and the cost function's approximate Hessian matrix, known as the Outer Product of Gradients (OPG) matrix. Furthermore, as the number of parameters (weights and biases) in deep learning usually is very large, we show how to reduce the quadratic space complexity by an efficient implementation based on approximate eigendecompositions.

cs.LG↗

arXiv2021

Epistemic Uncertainty Quantification in Deep Learning Classification by the Delta Method

The Delta method is a classical procedure for quantifying epistemic uncertainty in statistical models, but its direct application to deep neural networks is prevented by the large number of parameters $P$. We propose a low cost variant of the Delta method applicable to $L_2$-regularized deep neural networks based on the top $K$ eigenpairs of the Fisher information matrix. We address efficient computation of full-rank approximate eigendecompositions in terms of either the exact inverse Hessian, the inverse outer-products of gradients approximation or the so-called Sandwich estimator. Moreover, we provide a bound on the approximation error for the uncertainty of the predictive class probabilities. We observe that when the smallest eigenvalue of the Fisher information matrix is near the $L_2$-regularization rate, the approximation error is close to zero even when $K\ll P$. A demonstration of the methodology is presented using a TensorFlow implementation, and we show that meaningful rankings of images based on predictive uncertainty can be obtained for two LeNet-based neural networks using the MNIST and CIFAR-10 datasets. Further, we observe that false positives have on average a higher predictive epistemic uncertainty than true positives. This suggests that there is supplementing information in the uncertainty measure not captured by the classification alone.

cs.LG↗

arXiv2019

Relative persistent homology

The alpha complex efficiently computes persistent homology of a point cloud $X$ in Euclidean space when the dimension $d$ is low. Given a subset $A$ of $X$, relative persistent homology can be computed as the persistent homology of the relative \v{C}ech complex. But this is not computationally feasible for larger point clouds. The aim of this note is to present a method for efficient computation of relative persistent homology in low dimensional Euclidean space. We introduce the relative Delaunay \v{C}ech complex whose homology is the relative persistent homology. It can be constructed from the Delaunay complex of an embedding of the point clouds in $(d+1)$-dimensional Euclidean space.

math.AT↗

arXiv2019

Sparse Filtered Nerves

Given a point cloud $P$ in Euclidean space and a positive parameter $t$ we can consider the $t$-neighborhood $P^{t}$ of $P$ consisting of points at distance less than $t$ to $P$. Homology of $P^{t}$ gives information about components, holes, voids etc. in $P^{t}$. The idea of persistent homology is that it may happen that we are interested in some of holes in the spaces $P^t$ that are not detected simultaneously in homology for a single value of $t$, but where each of these holes is detected for $t$ in a wide range. When the dimension of the ambient Euclidean space is small, persistent homology is efficiently computed by the $α$-complex. For dimension bigger than three this becomes resource consuming. Don Sheehy discovered that there exists a filtered simplicial complex whose size depends linearly on the cardinality of $P$ and whose persistent homology is an approximation of the persistent homology of the filtered topological space $\{P^{t}\}_{t \ge 0}$. In this paper we pursue Sheehy's sparsification approach and give a more general approach to sparsification of filtered simplicial complexes computing the homology of filtered spaces of the form $\{P^{t}\}_{t \ge 0}$ and more generally to sparsification of filtered Dowker nerves. To our best knowledge, this is the first approach to sparsification of general Dowker nerves.

math.AT↗

arXiv2019

Sparse Nerves in Practice

Topological data analysis combines machine learning with methods from algebraic topology. Persistent homology, a method to characterize topological features occurring in data at multiple scales is of particular interest. A major obstacle to the wide-spread use of persistent homology is its computational complexity. In order to be able to calculate persistent homology of large datasets, a number of approximations can be applied in order to reduce its complexity. We propose algorithms for calculation of approximate sparse nerves for classes of Dowker dissimilarities including all finite Dowker dissimilarities and Dowker dissimilarities whose homology is Cech persistent homology. All other sparsification methods and software packages that we are aware of calculate persistent homology with either an additive or a multiplicative interleaving. In dowker_homology, we allow for any non-decreasing interleaving function $α$. We analyze the computational complexity of the algorithms and present some benchmarks. For Euclidean data in dimensions larger than three, the sizes of simplicial complexes we create are in general smaller than the ones created by SimBa. Especially when calculating persistent homology in higher homology dimensions, the differences can become substantial.

math.AT↗

arXiv2018

Divisive cover

The aim of this paper is to present a method for computation of persistent homology that performs well at large filtration values. To this end we introduce the concept of filtered covers. We show that the persistent homology of a bounded metric space obtained from the Čech complex is the persistent homology of the filtered nerve of the filtered Čech cover. Given a parameter $δ$ with $0 < δ\le 1$ we introduce the concept of a $δ$-filtered cover and show that its filtered nerve is interleaved with the Čech complex. Finally, we introduce a particular $δ$-filtered cover, the divisive cover. The special feature of the divisive cover is that it is constructed top-down. If we disregard fine scale structure and $X$ is a finite subspace of euclidean space, then we obtain a filtered simplicial complex whose size is bounded by an upper bound independent of the cardinality of $X$. The time needed to compute this filtered simplicial complex depends linearly on the cardinality of $X$.

math.AT↗