Related papers: Model comparison via simplicial complexes and pers…
Morse complexes and Morse-Smale complexes are topological descriptors popular in topology-based visualization. Comparing these complexes plays an important role in their applications in feature correspondences, feature tracking, symmetry…
Hypergraph is a topological model for networks. In order to study the topology of hypergraphs, the homology of the associated simplicial complexes and the embedded homology have been invented. In this paper, we give some algorithms to…
In this paper we introduce a statistic, the persistent homology transform (PHT), to model surfaces in $\mathbb{R}^3$ and shapes in $\mathbb{R}^2$. This statistic is a collection of persistence diagrams - multiscale topological summaries…
A method to apply and visualize persistent homology of time series is proposed. The method captures persistent features in space and time, in contrast to the existing procedures, where one usually chooses one while keeping the other fixed.…
This paper introduces an unsupervised method to estimate the class separability of text datasets from a topological point of view. Using persistent homology, we demonstrate how tracking the evolution of embedding manifolds during training…
The homological scaffold leverages persistent homology to construct a topologically sound summary of a weighted network. However, its crucial dependency on the choice of representative cycles hinders the ability to trace back global…
Many learning algorithms such as kernel machines, nearest neighbors, clustering, or anomaly detection, are based on the concept of 'distance' or 'similarity'. Before similarities are used for training an actual machine learning model, we…
We consider simplicial sets equipped with a notion of smallness, and observe that this slight "topological" extension of the "algebraic" simplicial language allows a concise reformulation of a number of classical notions in topology, e.g.…
Persistent homology is a popular data analysis technique that is used to capture the changing topology of a filtration associated with some simplicial complex $K$. These topological changes are summarized in persistence diagrams. We propose…
Many complex systems can be represented as networks, and the problem of network comparison is becoming increasingly relevant. There are many techniques for network comparison, from simply comparing network summary statistics to…
We produce a highly structured way of associating a simplicial category to a model category which improves on work of Dwyer and Kan and answers a question of Hovey. We show that model categories satisfying a certain axiom are Quillen…
In scientific inference problems, the underlying statistical modeling assumptions have a crucial impact on the end results. There exist, however, only a few automatic means for validating these fundamental modelling assumptions. The…
We propose a series of methods to represent the evolution of a field of science at different levels: namely micro, meso and macro levels. We use a previously introduced asymmetric measure of paradigmatic proximity between terms that enables…
Many real networks in social sciences, biological and biomedical sciences or computer science have an inherent structure of simplicial complexes reflecting many-body interactions. Therefore, to analyse topological and dynamical properties…
A well-defined distance on the parameter space is key to evaluating estimators, ensuring consistency, and building confidence sets. While there are typically standard distances to adopt in a continuous space, this is not the case for…
Compartmental epidemic models have been widely used for predicting the course of epidemics, from estimating the basic reproduction number to guiding intervention policies. Studies commonly acknowledge these models' assumptions but less…
We use the persistent homology method of topological data analysis and dimensional analysis techniques to study data of syntactic structures of world languages. We analyze relations between syntactic parameters in terms of dimensionality,…
We study properties of contiguity distance between simplicial maps. In particular, we show that simplicial versions of $LS$-category and topological complexity are particular cases of this more general notion.
Collective Adaptive Systems often consist of many heterogeneous components typically organised in groups. These entities interact with each other by adapting their behaviour to pursue individual or collective goals. In these systems, the…
Model-based clustering is widely-used in a variety of application areas. However, fundamental concerns remain about robustness. In particular, results can be sensitive to the choice of kernel representing the within-cluster data density.…