Related papers: Comparing persistence diagrams through complex vec…

Persistence diagrams, combining geometry and topology for an effective shape description used in pattern recognition, have already proven to be an effective tool for shape representation with respect to a certainfiltering function.…

It is known that for a variety of choices of metrics, including the standard bottleneck distance, the space of persistence diagrams admits geodesics. Typically these existence results produce geodesics that have the form of a convex…

The Persistent Homology Transform (PHT) summarizes a shape in $\mathbb{R}^m$ by collecting persistence diagrams obtained from linear height filtrations in all directions on $\mathbb{S}^{m-1}$. It enjoys strong theoretical guarantees,…

We redevelop persistent homology (topological persistence) from a categorical point of view. The main objects of study are diagrams, indexed by the poset of real numbers, in some target category. The set of such diagrams has an interleaving…

Persistence diagrams have been widely used to quantify the underlying features of filtered topological spaces in data visualization. In many applications, computing distances between diagrams is essential; however, computing these distances…

Throughout this paper, a persistence diagram ${\cal P}$ is composed of a set $P$ of planar points (each corresponding to a topological feature) above the line $Y=X$, as well as the line $Y=X$ itself, i.e., ${\cal P}=P\cup\{(x,y)|y=x\}$.…

Topological Data Analysis methods can be useful for classification and clustering tasks in many different fields as they can provide two dimensional persistence diagrams that summarize important information about the shape of potentially…

In topological data analysis (TDA), persistence diagrams have been a succesful tool. To compare them, Wasserstein and Bottleneck distances are commonly used. We address the shortcomings of these metrics and show a way to investigate them in…

One of the main reasons for topological persistence being useful in data analysis is that it is backed up by a stability (isometry) property: persistence diagrams of $1$-parameter persistence modules are stable in the sense that the…

Recent years have witnessed a tremendous growth using topological summaries, especially the persistence diagrams (encoding the so-called persistent homology) for analyzing complex shapes. Intuitively, persistent homology maps a potentially…

Given two point sets in the plane, we study the minimization of the bottleneck distance between a point set B and an equally-sized subset of a point set A under translations. We relate this problem to a Voronoi-type diagram and derive…

Computation of the interleaving distance between persistence modules is a central task in topological data analysis. For $1$-parameter persistence modules, thanks to the isometry theorem, this can be done by computing the bottleneck…

Exploiting geometric structure to improve the asymptotic complexity of discrete assignment problems is a well-studied subject. In contrast, the practical advantages of using geometry for such problems have not been explored. We implement…

Distance-preserving mappings (DPMs) are mappings from the set of all q-ary vectors of a fixed length to the set of permutations of the same or longer length such that every two distinct vectors are mapped to permutations with the same or…

Persistent homology has become an important tool for extracting geometric and topological features from data, whose multi-scale features are summarized in a persistence diagram. From a statistical perspective, however, persistence diagrams…

This paper explores persistence modules for circle-valued functions, presenting a new extension of the interleaving and bottleneck distances in this setting. We propose a natural generalisation of barcodes in terms of arcs on a geometric…

An important application of distance geometry to biochemistry studies the embeddings of the vertices of a weighted graph in the three-dimensional Euclidean space such that the edge weights are equal to the Euclidean distances between…

The problem of finding the distance from a given $n \times n$ matrix polynomial of degree $k$ to the set of matrix polynomials having the elementary divisor $(\lambda-\lambda_0)^j, \, j \geqslant r,$ for a fixed scalar $\lambda_0$ and $2…

In multi-parameter persistence, the matching distance is defined as the supremum of weighted bottleneck distances on the barcodes given by the restriction of persistence modules to lines with a positive slope. In the case of finitely…

Behaviour distances to measure the resemblance of two states in a (nondeterministic) fuzzy transition system have been proposed recently in the literature. Such a distance, defined as a pseudo-ultrametric over the state space of the model,…