Related papers: The Persistent Homotopy Type Distance
In this paper we present a novel methodology based on a topological entropy, the so-called persistent entropy, for addressing the comparison between discrete piecewise linear functions. The comparison is certified by the stability theorem…
Many datasets can be viewed as a noisy sampling of an underlying space, and tools from topological data analysis can characterize this structure for the purpose of knowledge discovery. One such tool is persistent homology, which provides a…
This paper proposes an algorithm that decides if two simply connected spaces represented by finite simplicial sets of finite $k$-type and finite dimension $d$ are homotopy equivalent. If the spaces are homotopy equivalent, the algorithm…
We show that basic homotopical notions such as homotopy sets and groups, connected and truncated maps, cellular constructions and skeleta, etc., extend to the setting of $(\infty,\infty)$-categories, as well as to presentable categories…
The interleaving distance is arguably the most prominent distance measure in topological data analysis. In this paper, we provide bounds on the computational complexity of determining the interleaving distance in several settings. We show…
We take a different look at the problem of testing the independence of two metric-space-valued random variables using the distance correlation. Instead of testing if the distance correlation vanishes exactly, we are interested in the…
In this work, we introduce a notion of reachability entropy to characterize the smallest data rate which is sufficient enough to enforce reach-while-stay specification. We also define data rates of coder-controllers that can enforce this…
We introduce the concept of homotopy iterators for performing polynomial homotopy continuation tasks in a memory efficient manner. The main idea is to push forward an iterator for the start solutions of a homotopy via the function which…
We define a Lorentzian distance function on the group of contactomorphisms of a closed contact manifold. This distance function is continuous with respect to the Hofer norm on the group of contactomorphisms defined by Shelukhin and finite…
To quantify the fundamental evolution of time-varying networks, and detect abnormal behavior, one needs a notion of temporal difference that captures significant organizational changes between two successive instants. In this work, we…
The paper introduces a class of distances for linear behaviors over finite time horizons. These distances allow for comparisons between finite-horizon linear behaviors represented by matrices of possibly different dimensions. They remain…
Persistent homology (PH) is a recently developed theory in the field of algebraic topology to study shapes of datasets. It is an effective data analysis tool that is robust to noise and has been widely applied. We demonstrate a general…
The aim of this paper is to introduce a new class of softly normal called softly $\pi g\widehat{D}$ -normality by using $\pi g\widehat{D}$ -open sets and obtained several properties of such a space. We discuss many properties of this new…
Persistent homology (PH) is a method used in topological data analysis (TDA) to study qualitative features of data that persist across multiple scales. It is robust to perturbations of input data, independent of dimensions and coordinates,…
Let D be a division algebra over a base field k. The homological transcendence degree of D, denoted by Htr D, is defined to be the injective dimension of the enveloping algebra of D. We show that Htr has several useful properties which the…
Homotopy Type Theory is a new field of mathematics based on the surprising and elegant correspondence between Martin-Lofs constructive type theory and abstract homotopy theory. We have a powerful interplay between these disciplines - we can…
The literature postulates that the dynamic time warping (dtw) distance can cope with temporal variations but stores and processes time series in a form as if the dtw-distance cannot cope with such variations. To address this inconsistency,…
Higher-order proximity (HOP) is fundamental for most network embedding methods due to its significant effects on the quality of node embedding and performance on downstream network analysis tasks. Most existing HOP definitions are based on…
This paper is a survey of persistent homology, primarily as it is used in topological data analysis. It includes the theory of persistence modules, as well as stability theorems for persistence barcodes, generalized persistence,…
Motivated by the approach of random linear codes, a new distance in the vector space over a finite field is defined as the logarithm of the "surface area" of a Hamming ball with radius being the corresponding Hamming distance. It is named…