Related papers: Over then Under Tangles
Overlap is a common phenomenon seen when structural components of a digital object are neither disjoint nor nested inside each other. Overlapping components resist reduction to a structural hierarchy, and tree-based indexing and query…
Higher-dimensional category theory is the study of n-categories, operads, braided monoidal categories, and other such exotic structures. Although it can be treated purely as an algebraic subject, it is inherently topological in nature: the…
A tangled-diagram over $[n]=\{1,...,n\}$ is a graph of degree less than two whose vertices $1,...,n$ are arranged in a horizontal line and whose arcs are drawn in the upper halfplane with a particular notion of crossings and nestings.…
Optimal Transport (OT) has established itself as a robust framework for quantifying differences between distributions, with applications that span fields such as machine learning, data science, and computer vision. This paper offers a…
This paper studies the underlying combinatorial structure of a class of object rearrangement problems, which appear frequently in applications. The problems involve multiple, similar-geometry objects placed on a flat, horizontal surface,…
A clustering algorithm partitions a set of data points into smaller sets (clusters) such that each subset is more tightly packed than the whole. Many approaches to clustering translate the vector data into a graph with edges reflecting a…
Structurally stable (rough) flows on surfaces have only finitely many singularities and finitely many closed orbits, all of which are hyperbolic, and they have no trajectories joining saddle points. The violation of the last property leads…
Deep learning-based approaches have achieved significant improvements on public video anomaly datasets, but often do not perform well in real-world applications. This paper addresses two issues: the lack of labeled data and the difficulty…
We begin with a short presentation of the basic concepts related to Lie groupoids and Lie algebroids, but the main part of this paper deals with Lie algebroids. A Lie algebroid over a manifold is a vector bundle over that manifold whose…
Introduced by Papadimitriou and Yannakakis in 1989, layered graph traversal is a central problem in online algorithms and mobile computing that has been studied for several decades, and which now is essentially resolved in its original…
We study tilting complexes over preprojective algebras of Dynkin type. We classify all tilting complexes by giving a bijection between tilting complexes and the braid group of the corresponding folded graph. In particular, we determine the…
The topological model for quantum computation is an inherently fault-tolerant model built on anyons in topological phases of matter. A key role is played by the braid group, and in this survey we focus on a selection of ways that the…
Hidden symmetries are the backbone of Integrable two-dimensional theories. They provide classical solutions of higher dimensional models as well, they seem to survive partially quantisation and their discrete remnants in M-theory called…
A general notion of operad is given, which includes as instances, the operads originally conceived to study loop spaces, as well as the higher operads that arise in the globular approach to higher dimensional algebra. In the framework of…
Graph convolutions have gained popularity due to their ability to efficiently operate on data with an irregular geometric structure. However, graph convolutions cause over-smoothing, which refers to representations becoming more similar…
Distribution shifts on graphs -- the data distribution discrepancies between training and testing a graph machine learning model, are often ubiquitous and unavoidable in real-world scenarios. Such shifts may severely deteriorate the…
${\cal U}$ntil now the representation (i.e. plotting) of curve in Parallel Coordinates is constructed from the point $\leftrightarrow$ line duality. The result is a ``line-curve'' which is seen as the envelope of it's tangents. Usually this…
The enduring legacy of Euclidean geometry underpins classical machine learning, which, for decades, has been primarily developed for data lying in Euclidean space. Yet, modern machine learning increasingly encounters richly structured data…
We describe all possible bimodal over-twist patterns. In particular, we give an algorithm allowing one to determine what the left endpoint of the over-rotation interval of a given bimodal map is. We then define a new class of polymodal…
A graph is beyond-planar if it can be drawn in the plane with a specific restriction on crossings. Several types of beyond-planar graphs have been investigated, such as k-planar if every edge is crossed at most k times and RAC if edges can…