Related papers: Interleaving distances from height-difference func…
Ladders of recollements of abelian categories are introduced, and used to address three general problems. Ladders of a certain height allow to construct recollements of triangulated categories, involving derived categories and singularity…
In this paper an improved method for searching good interleavers from a certain set is proposed. The first few terms, corresponding to maximum distance of approximately 40 of the distance spectra, for turbo codes using these interleavers…
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.…
In the Stanley lattice defined on Dyck paths of size $n$, cover relations are obtained by replacing a valley $DU$ by a peak $UD$. We investigate a greedy version of this lattice, first introduced by Chenevi\`ere, where cover relations…
Persistent homology allows us to create topological summaries of complex data. In order to analyse these statistically, we need to choose a topological summary and a relevant metric space in which this topological summary exists. While…
We introduce lifespan functors, which are endofunctors on the category of persistence modules that filter out intervals from barcodes according to their boundedness properties. They can be used to classify injective and projective objects…
For safety and robustness of AI systems, we introduce topological parallax as a theoretical and computational tool that compares a trained model to a reference dataset to determine whether they have similar multiscale geometric structure.…
Proposing scoring functions to effectively understand, analyze and learn various properties of high dimensional hidden representations of large-scale transformer models like BERT can be a challenging task. In this work, we explore a new…
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…
In Persistent Homology and Topology, filtrations are usually given by introducing an ordered collection of sets or a continuous function from a topological space to $\R^n$. A natural question arises, whether these approaches are equivalent…
In this paper, we investigate a discrete version of the homotopic distance between two $s$-Lipschitz maps for $s \geq 0$. This distance is defined by specifying a step length $r$ to which some homotopy relation corresponds. In spaces with a…
The goal of this paper is to learn dense 3D shape correspondence for topology-varying objects in an unsupervised manner. Conventional implicit functions estimate the occupancy of a 3D point given a shape latent code. Instead, our novel…
The pose of a rigid object is usually regarded as a rigid transformation, described by a translation and a rotation. However, equating the pose space with the space of rigid transformations is in general abusive, as it does not account for…
This article concerns a systemic manifestation of small scale interfacial heterogeneities in large scale quantities of interest to a variety of diverse applications spanning the earth, biological and ecological sciences. Beginning with…
This paper studies the properties of a new lower bound for the natural pseudo-distance. The natural pseudo-distance is a dissimilarity measure between shapes, where a shape is viewed as a topological space endowed with a real-valued…
Comparison between multidimensional persistent Betti numbers is often based on the multidimensional matching distance. While this metric is rather simple to define and compute by considering a suitable family of filtering functions…
Diffusion, a fundamental internal mechanism emerging in many physical processes, describes the interaction among different objects. In many learning tasks with limited training samples, the diffusion connects the labeled and unlabeled data…
We introduce posets with interfaces (iposets) and generalise their standard serial composition to a new gluing composition. In the partial order semantics of concurrency, interfaces and gluing allow modelling events that extend in time and…
This paper defines a distance function that measures the dissimilarity between planar geometric figures formed with straight lines. This function can in turn be used in partial matching of different geometric figures. For a given pair of…
We develop the theoretical foundations of a generalized Gromov-Hausdorff distance between functions on networks that has recently been applied to various subfields of topological data analysis and optimal transport. These functional…