Related papers: Combinatorial Persistent Homology Transform
Persistent homology captures the evolution of topological features of a model as a parameter changes. The most commonly used summary statistics of persistent homology are the barcode and the persistence diagram. Another summary statistic,…
For a fixed $N$, we analyze the space of all sequences $z=(z_1,\dots,z_N)$, approximating a continuous function on the circle, with a given persistence diagram $P$, and show that the typical components of this space are homotopy equivalent…
Topological data analysis involves the statistical characterization of the shape of data. Persistent homology is a primary tool of topological data analysis, which can be used to analyze topological features and perform statistical…
Persistent homology (PH) is one of the most popular methods in Topological Data Analysis. Even though PH has been used in many different types of applications, the reasons behind its success remain elusive; in particular, it is not known…
The study of combinatorial properties of mathematical objects is a very important research field and continued fractions have been deeply studied in this sense. However, multidimensional continued fractions, which are a generalization…
We use Topological Data Analysis tools for studying the inner organization of cells in segmented images of epithelial tissues. More specifically, for each segmented image, we compute different persistence barcodes, which codify lifetime of…
Persistence homology is a tool used to measure topological features that are present in data sets and functions. Persistence pairs births and deaths of these features as we iterate through the sublevel sets of the data or function of…
In this work, we explore links between natural homology and persistent homology for the classification of directed spaces. The former is an algebraic invariant of directed spaces, a semantic model of concurrent programs. The latter was…
We define and study a class of finite topological spaces, which model the cell structure of a space obtained by gluing finitely many Euclidean convex polyhedral cells along congruent faces. We call these finite topological spaces,…
The computational cost of persistent homology is often dominated by the growth of the underlying simplicial filtrations. Many different filtrations exist, each with its own assumptions and trade-offs, but all face some form of this growth…
An algorithm is presented that constructs an acyclic partial matching on the cells of a given simplicial complex from a vector-valued function defined on the vertices and extended to each simplex by taking the least common upper bound of…
A flat-injective presentation of a multiparameter persistence module $M$ characterizes $M$ as the image of a morphism from a flat to an injective persistence module. Like flat or injective presentations, flat-injective presentations can be…
We reformulate the persistent (co)homology of simplicial filtrations, viewed from a more algebraic setting, namely as the (co)homology of a chain complex of graded modules over polynomial ring $K[t]$. We also define persistent (co)homology…
Over the past two decades, topological data analysis has emerged as a field of applied mathematics with new applications and algorithmic developments appearing rapidly. Two fundamental computations in this field are persistent homology and…
Persistent homology is currently one of the more widely known tools from computational topology and topological data analysis. We present in this note a brief survey on the evolution of the subject. The goal is to highlight the main ideas,…
In this paper we examine the existence of bicomplexified inverse Fourier transform as an extension of its complexified inverse version within the region of convergence of bicomplex Fourier transform. In this paper we use the idempotent…
This paper studies the homotopy-type of bi-filtrations of compact manifolds induced as the pre-image of filtrations of the plane for generic smooth functions f : M --> R^2. The primary goal of the paper is to allow for a simple description…
Existing interpretability methods for Large Language Models (LLMs) predominantly capture linear directions or isolated features. This overlooks the high-dimensional, relational, and nonlinear geometry of model representations. We apply…
Vietoris-Rips and degree Rips complexes are represented as homotopy types by their underlying posets of simplices, and basic homotopy stability theorems are recast in these terms. These homotopy types are viewed as systems (or functors),…
Persistent homology is a multiscale method for analyzing the shape of sets and functions from point cloud data arising from an unknown distribution supported on those sets. When the size of the sample is large, direct computation of the…