Related papers: Measure Theoretic Reeb Graphs and Reeb Spaces
Whether comparing networks to each other or to random expectation, measuring dissimilarity is essential to understanding the complex phenomena under study. However, determining the structural dissimilarity between networks is an ill-defined…
Persistence diagrams, the most common descriptors of Topological Data Analysis, encode topological properties of data and have already proved pivotal in many different applications of data science. However, since the (metric) space of…
Fold maps are higher dimensional versions of Morse functions and fundamental and important tools in studying algebraic and differential topological properties of manifolds: as the theory established by Morse and the higher dimensional…
In this paper, we defined three kinds of measures depending on the given finite directed graphs. For the given finite directed graph, we can construct the free semigroupoid, the diagram set and the reduced diagram set, as algebraic…
Metric graphs are special types of metric spaces used to model and represent simple, ubiquitous, geometric relations in data such as biological networks, social networks, and road networks. We are interested in giving a qualitative…
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 show that the collection of regular Borel measures on a second-countable locally compact Hausdorff space has the structure of a sheaf. With this we give an alternate description of the pullback of a regular Borel measure along a local…
This paper investigates spaces equipped with a family of metric-like functions satisfying certain axioms. These functions provide a unified framework for defining topology, uniformity, and diffeology. The framework is based on a family of…
In a recent paper, two multi-representations for the measurable sets in a computable measure space have been introduced, which prove to be topologically complete w.r.t. certain topological properties. In this contribution, we show them…
The problem of graph learning concerns the construction of an explicit topological structure revealing the relationship between nodes representing data entities, which plays an increasingly important role in the success of many graph-based…
We consider different notions of equivalence for Morse functions on the sphere in the context of persistent homology, and introduce new invariants to study these equivalence classes. These new invariants are as simple, but more discerning…
Interactions and relations between objects may be pairwise or higher-order in nature, and so network-valued data are ubiquitous in the real world. The "space of networks", however, has a complex structure that cannot be adequately described…
A natural representation of random graphs is the random measure. The collection of product random measures, their transformations, and non-negative test functions forms a general representation of the collection of non-negative weighted…
We study the metric structure of walks on graphs, understood as Lipschitz sequences. To this end, a weighted metric is introduced to handle sequences, enabling the definition of distances between walks based on stepwise vertex distances and…
Classes of branched surfaces extend the classes of surfaces or 2-dimensional manifolds satisfying suitable properties and defined in various manners. Reeb spaces of smooth maps of suitable classes into surfaces whose codimensions are…
We study the induced measure obtained from a 1-step Markov measure, supported by a topological Markov chain, after the mapping of the original alphabet onto another one. We give sufficient conditions for the induced measure to be a Gibbs…
In this thesis, the main objects of study are probability measures on the isomorphism classes of countable, connected rooted graphs. An important class of such measures is formed by unimodular measures, which satisfy a certain equation,…
Learning faithful graph representations as sets of vertex embeddings has become a fundamental intermediary step in a wide range of machine learning applications. The quality of the embeddings is usually determined by how well the geometry…
Persistence diagrams are important descriptors in Topological Data Analysis. Due to the nonlinearity of the space of persistence diagrams equipped with their {\em diagram distances}, most of the recent attempts at using persistence diagrams…
Persistence diagrams are common descriptors of the topological structure of data appearing in various classification and regression tasks. They can be generalized to Radon measures supported on the birth-death plane and endowed with an…