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Mikael Vejdemo-Johansson

Publications and source records attributed to Mikael Vejdemo-Johansson.

At least 19 recordsLinked to original sources
arXiv2026

Incomplete Open Platonic Solids

Sol LeWitt famously enumerated all the incomplete open cubes, finding 122 of these connected, non-planar subsets of the edges of the cube. Since then, while several projects have revisited the cube enumeration, no such enumeration has been published for any other interesting solid. In this paper we present work on enumerating all the incomplete open platonic solids, finding 6 tetrahedra, 122 cubes (just like LeWitt), 185 octahedra, 2\,423\,206 dodecahedra and 16\,096\,166 icosahedra.

math.CO↗
arXiv2025

$k$-means considered harmful: On arbitrary topological changes in Mapper complexes

The Mapper construction is one of the most widespread tools from Topological Data Analysis. There is an unfortunate trend as the construction has gained traction to use clustering methods with properties that end up distorting any analysis results from the construction. In this paper we will see a few ways in which widespread choices of clustering algorithms have arbitrarily large distortions of the features visible in the final Mapper complex.

cs.CG↗
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↗
arXiv2023

Topological Learning for Motion Data via Mixed Coordinates

Topology can extract the structural information in a dataset efficiently. In this paper, we attempt to incorporate topological information into a multiple output Gaussian process model for transfer learning purposes. To achieve this goal, we extend the framework of circular coordinates into a novel framework of mixed valued coordinates to take linear trends in the time series into consideration. One of the major challenges to learn from multiple time series effectively via a multiple output Gaussian process model is constructing a functional kernel. We propose to use topologically induced clustering to construct a cluster based kernel in a multiple output Gaussian process model. This kernel not only incorporates the topological structural information, but also allows us to put forward a unified framework using topological information in time and motion series.

cs.LG↗
arXiv2022

An improved algorithm for Generalized Čech complex construction

In this paper, we present an algorithm that computes the generalized Čech complex for a finite set of disks where each may have a different radius in 2D space. An extension of this algorithm is also proposed for a set of balls in 3D space with different radius. To compute a $k$-simplex, we leverage the computation performed in the round of $(k-1)$-simplices such that we can reduce the number of potential candidates to verify to improve the efficiency. An efficient verification method is proposed to confirm if a $k$-simplex can be constructed on the basis of the $(k-1)$-simplices. We demonstrate the performance with a comparison to some closely related algorithms.

cs.CG↗
arXiv2022

Multiple testing with persistent homology

In this paper we propose a computationally efficient multiple hypothesis testing procedure for persistent homology. The computational efficiency of our procedure is based on the observation that one can empirically simulate a null distribution that is universal across many hypothesis testing applications involving persistence homology. Our observation suggests that one can simulate the null distribution efficiently based on a small number of summaries of the collected data and use this null in the same way that p-value tables were used in classical statistics. To illustrate the efficiency and utility of the null distribution we provide procedures for rejecting acyclicity with both control of the Family-Wise Error Rate (FWER) and the False Discovery Rate (FDR). We will argue that the empirical null we propose is very general conditional on a few summaries of the data based on simulations and limit theorems for persistent homology for point processes.

cs.CG↗
arXiv2021

Generalized Penalty for Circular Coordinate Representation

Topological Data Analysis (TDA) provides novel approaches that allow us to analyze the geometrical shapes and topological structures of a dataset. As one important application, TDA can be used for data visualization and dimension reduction. We follow the framework of circular coordinate representation, which allows us to perform dimension reduction and visualization for high-dimensional datasets on a torus using persistent cohomology. In this paper, we propose a method to adapt the circular coordinate framework to take into account the roughness of circular coordinates in change-point and high-dimensional applications. We use a generalized penalty function instead of an $L_{2}$ penalty in the traditional circular coordinate algorithm. We provide simulation experiments and real data analysis to support our claim that circular coordinates with generalized penalty will detect the change in high-dimensional datasets under different sampling schemes while preserving the topological structures.

stat.ML↗
arXiv2019

Algorithms in $A_\infty$-algebras

Building on Kadeishvili's original theorem inducing $A_\infty$-algebra structures on the homology of dg-algebras, several directions of algorithmic research in $A_\infty$-algebras have been pursued. In this paper we will survey work done on calculating explicit $A_\infty$-algebra structures from homotopy retractions; in group cohomology; and in persistent homology.

math.KT↗
arXiv2018

Certified Mapper: Repeated testing for acyclicity and obstructions to the nerve lemma

The Mapper algorithm does not include a check for whether the cover produced conforms to the requirements of the nerve lemma. To perform a check for obstructions to the nerve lemma, statistical considerations of multiple testing quickly arise. In this paper, we propose several statistical approaches to finding obstructions: through a persistent nerve lemma, through simulation testing, and using a parametric refinement of simulation tests. We suggest Certified Mapper -- a method built from these approaches to generate certificates of non-obstruction, or identify specific obstructions to the nerve lemma -- and we give recommendations for which statistical approaches are most appropriate for the task.

math.AT↗
arXiv2018

Fibres of Failure: Classifying errors in predictive processes

We describe Fibres of Failure (FiFa), a method to classify failure modes of predictive processes using the Mapper algorithm from Topological Data Analysis. Our method uses Mapper to build a graph model of input data stratified by prediction error. Groupings found in high-error regions of the Mapper model then provide distinct failure modes of the predictive process. We demonstrate FiFa on misclassifications of MNIST images with added noise, and demonstrate two ways to use the failure mode classification: either to produce a correction layer that adjusts predictions by similarity to the failure modes; or to inspect members of the failure modes to illustrate and investigate what characterizes each failure mode.

cs.CV↗
arXiv2017

Weighted growth functions of automatic groups

The growth function is the generating function for sizes of spheres around the identity in Cayley graphs of groups. We present a novel method to calculate growth functions for automatic groups with normal form recognizing automata that recognize a single normal form for each group element, and are at most context free in complexity: context free grammars can be translated into algebraic systems of equations, whose solutions represent generating functions of their corresponding non-terminal symbols. This approach allows us to seamlessly introduce weightings on the growth function: assign different or even distinct weights to each of the generators in an underlying presentation, such that this weighting is reflected in the growth function. We recover known growth functions for small braid groups, and calculate growth functions that weight each generator in an automatic presentation of the braid groups according to their lengths in braid generators.

math.GR↗
arXiv2015

More ties than we thought

We extend the existing enumeration of neck tie-knots to include tie-knots with a textured front, tied with the narrow end of a tie. These tie-knots have gained popularity in recent years, based on reconstructions of a costume detail from The Matrix Reloaded, and are explicitly ruled out in the enumeration by Fink and Mao (2000). We show that the relaxed tie-knot description language that comprehensively describes these extended tie-knot classes is context free. It has a regular sub-language that covers all the knots that originally inspired the work. From the full language, we enumerate 266 682 distinct tie-knots that seem tie-able with a normal neck-tie. Out of these 266 682, we also enumerate 24 882 tie-knots that belong to the regular sub-language.

cs.FL↗
arXiv2014

Computing minimum area homologies

Calculating and categorizing the similarity of curves is a fundamental problem which has generated much recent interest. However, to date there are no implementations of these algorithms for curves on surfaces with provable guarantees on the quality of the measure. In this paper, we present a similarity measure for any two cycles that are homologous, where we calculate the minimum area of any homology (or connected bounding chain) between the two cycles. The minimum area homology exists for broader classes of cycles than previous measures which are based on homotopy. It is also much easier to compute than previously defined measures, yielding an efficient implementation that is based on linear algebra tools. We demonstrate our algorithm on a range of inputs, showing examples which highlight the feasibility of this similarity measure.

cs.CG↗
arXiv2014

Aspects of an internal logic for persistence

The foundational character of certain algebraic structures as Boolean algebras and Heyting algebras is rooted in their potential to model classical and constructive logic, respectively. In this paper we discuss the contributions of algebraic logic to the study of persistence based on a new operation on the ordered structure of the input diagram of vector spaces and linear maps given by a filtration. Within the context of persistence theory, we give an analysis of the underlying algebra, derive universal properties and discuss new applications. We highlight the definition of the implication operation within this construction, as well as interpret its meaning within persistent homology, multidimensional persistence and zig-zag persistence.

math.RA↗
arXiv2014

Automatic recognition and tagging of topologically different regimes in dynamical systems

Complex systems are commonly modeled using nonlinear dynamical systems. These models are often high-dimensional and chaotic. An important goal in studying physical systems through the lens of mathematical models is to determine when the system undergoes changes in qualitative behavior. A detailed description of the dynamics can be difficult or impossible to obtain for high-dimensional and chaotic systems. Therefore, a more sensible goal is to recognize and mark transitions of a system between qualitatively different regimes of behavior. In practice, one is interested in developing techniques for detection of such transitions from sparse observations, possibly contaminated by noise. In this paper we develop a framework to accurately tag different regimes of complex systems based on topological features. In particular, our framework works with a high degree of success in picking out a cyclically orbiting regime from a stationary equilibrium regime in high-dimensional stochastic dynamical systems.

cs.CG↗
arXiv2013

Sketches of a platypus: persistent homology and its algebraic foundations

The subject of persistent homology has vitalized applications of algebraic topology to point cloud data and to application fields far outside the realm of pure mathematics. The area has seen several fundamentally important results that are rooted in choosing a particular algebraic foundational theory to describe persistent homology, and applying results from that theory to prove useful and important results. In this survey paper, we shall examine the various choices in use, and what they allow us to prove. We shall also discuss the inherent differences between the choices people use, and speculate on potential directions of research to resolve these differences.

math.AT↗
arXiv2013

Persistence modules: Algebra and algorithms

Persistent homology was shown by Carlsson and Zomorodian to be homology of graded chain complexes with coefficients in the graded ring $\kk[t]$. As such, the behavior of persistence modules -- graded modules over $\kk[t]$ is an important part in the analysis and computation of persistent homology. In this paper we present a number of facts about persistence modules; ranging from the well-known but under-utilized to the reconstruction of techniques to work in a purely algebraic approach to persistent homology. In particular, the results we present give concrete algorithms to compute the persistent homology of a simplicial complex with torsion in the chain complex.

cs.CG↗