Related papers: Probabilistic Analysis of Multiparameter Persisten…
In this paper, we study pointwise finite-dimensional (p.f.d.) $2$-parameter persistence modules where each module admits a finite convex isotopy subdivision. We show that a p.f.d. $2$-parameter persistence module $M$ (with a finite convex…
In this paper we explain how to convert discrete invariants into stable ones via what we call hierarchical stabilization. We illustrate this process by constructing stable invariants for multi-parameter persistence modules with respect to…
Persistence modules are representations of products of totally ordered sets in the category of vector spaces. They appear naturally in the representation theory of algebras, but in recent years they have also found applications in other…
The persistence barcode (equivalently, the persistence diagram), which can be obtained from the interval decomposition of a persistence module, plays a pivotal role in applications of persistent homology. For multi-parameter persistent…
When working with (multi-parameter) persistence modules, one usually makes some type of tameness assumption in order to obtain better control over their algebraic behavior. One such notion is Ezra Millers notion of finite encodability,…
In topological data analysis persistence modules are used to distinguish the legitimate topological features of a finite data set from noise. Interleavings between persistence modules feature prominantly in the analysis. One can show that…
The set of all permutations, ordered by pattern containment, forms a poset. This paper presents the first explicit major results on the topology of intervals in this poset. We show that almost all (open) intervals in this poset have a…
Zeckendorf's theorem states that every positive integer can be uniquely decomposed as a sum of nonconsecutive Fibonacci numbers. The distribution of the number of summands converges to a Gaussian, and the individual measures on gaps between…
We propose a functorial framework for persistent homology based on finite topological spaces and their associated posets. Starting from a finite metric space, we associate a filtration of finite topologies whose structure maps are…
In this work, we present a generalization of extended persistent homology to filtrations of graded sub-groups by defining relative homology in this setting. Our work provides a more comprehensive and flexible approach to get an algebraic…
A (fragment of a) process algebra satisfies unique parallel decomposition if the definable behaviours admit a unique decomposition into indecomposable parallel components. In this paper we prove that finite processes of the pi-calculus,…
Finite mixture models are statistical models which appear in many problems in statistics and machine learning. In such models it is assumed that data are drawn from random probability measures, called mixture components, which are…
We address the problem of estimating multiple modes of a multivariate density using persistent homology, a central tool in Topological Data Analysis. We introduce a method based on the preliminary estimation of the $H_0$-persistence diagram…
We study the indecomposable summands of the permutation module obtained by inducing the trivial $\mathbb{F}(S_a\wr S_n)$-module to the full symmetric group $S_{an}$ for any field $\mathbb{F}$ of odd prime characteristic $p$ such that…
Probabilistic inference procedures are usually coded painstakingly from scratch, for each target model and each inference algorithm. We reduce this effort by generating inference procedures from models automatically. We make this code…
Persistent homology is a popular and useful tool for analysing finite metric spaces, revealing features that can be used to distinguish sets of unlabeled points and as input into machine learning pipelines. The famous stability theorem of…
An important problem in the field of Topological Data Analysis is defining topological summaries which can be combined with traditional data analytic tools. In recent work Bubenik introduced the persistence landscape, a stable…
In this article, we give some conditions on the structure of an unstable module, which are satisfied whenever this module is the reduced cohomology of a space or a spectrum. First, we study the structure of the sub-modules of…
We create mechanically stable (MS) packings of bidisperse disks using an algorithm in which we successively grow or shrink soft repulsive disks followed by energy minimization until the overlaps are vanishingly small. We focus on small…
Random probabilities are a key component to many nonparametric methods in Statistics and Machine Learning. To quantify comparisons between different laws of random probabilities several works are starting to use the elegant Wasserstein over…