Related papers: Interleaving distances from height-difference func…

Difference sets and their generalisations to difference families arise from the study of designs and many other applications. Here we give a brief survey of some of these applications, noting in particular the diverse definitions of…

We define two notions. The first one is a $rank\ compression\ system$ $\xi$ for a finite poset $\mathbf{P}$ that assigns each interval subposet $I$ to an order-preserving map $\xi_I \colon I^{\xi} \to \mathbf{P}$ satisfying some conditions,…

Transit functions were introduced as models of betweenness on undirected structures. Here we introduce directed transit function as the directed analogue on directed structures such as posets and directed graphs. We first show that…

In this paper we study a new metric for comparing Betti numbers functions in bidimensional persistent homology, based on coherent matchings, i.e. families of matchings that vary in a continuous way. We prove some new results about this…

Support vector machines (SVMs) naturally embody sparseness due to their use of hinge loss functions. However, SVMs can not directly estimate conditional class probabilities. In this paper we propose and study a family of coherence…

Motivated both by theoretical and practical considerations in topological data analysis, we generalize the $p$-Wasserstein distance on barcodes to multiparameter persistence modules. For each $p\in [1,\infty]$, we in fact introduce two such…

Define an expansion poset to be the poset of monomials of a cluster variable attached to an arc in a polygon, where each monomial is represented by the corresponding combinatorial object from some fixed combinatorial cluster expansion…

From the work of Bauer and Lesnick, it is known that there is no functor from the category of pointwise finite-dimensional persistence modules to the category of barcodes and overlap matchings. In this work, we introduce sub-barcodes and…

In this paper, we introduce a new class of implicit function to prove common fixed point theorems in fuzzy metric space. Moreover we define a new altering distance in terms of integral and utilize the same to deduce integral type…

We develop the notion of a "filtered cospan" as an algebraic object that stands in the same relation to interlevel persistence modules as filtered chain complexes stand with respect to sublevel persistence modules. This relation is…

In general, representations of interval orders may use an arbitrary set of interval lengths. We can define subclasses of interval orders by restricting the allowable lengths of intervals. Motivated by a recent paper of Keller, Trenk, and…

We present a method for learning an embedding that places images of humans in similar poses nearby. This embedding can be used as a direct method of comparing images based on human pose, avoiding potential challenges of estimating body…

The interval poset of a permutation is the set of intervals of a permutation, ordered with respect to inclusion. It has been introduced and studied recently in [B. Tenner, arXiv:2007.06142]. We study this poset from the perspective of the…

Persistent homology is a fundamental tool in Topological Data Analysis. The associated algebraic structure is the persistence module, a sequence of vector spaces connected by linear maps. Persistence modules admit a complete and…

Unsupervised embeddings are fundamental to numerous machine learning applications, yet their evaluation remains a challenging task. Traditional assessment methods often rely on extrinsic variables, such as performance in downstream tasks,…

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…

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 consider the setting of Reeb graphs of piecewise linear functions and study distances between them that are stable, meaning that functions which are similar in the supremum norm ought to have similar Reeb graphs. We define an edit…

This paper introduces the induced matching distance, a novel topological metric designed to compare discrete structures represented by a symmetric non-negative function. We apply this notion to analyze agent trajectories over time. We use…

Understanding the decision-making processes of large language models is critical given their widespread applications. To achieve this, we aim to connect a formal mathematical framework - zigzag persistence from topological data analysis -…