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In this paper, we study metric trees, without any finiteness restrictions. For subsets of such trees, a condition that guarantees that the Hausdorff and Gromov--Hausdorff distances from the subset to the entire metric tree are the same is…
What is the shortest path between two data points lying in a high-dimensional space? While the answer is trivial in Euclidean geometry, it becomes significantly more complex when the data lies on a curved manifold -- requiring a Riemannian…
We establish that many fundamental concepts and techniques in quantum field theory and collider physics can be naturally understood and unified through a simple new geometric language. The idea is to equip the space of collider events with…
The structural analysis of shape boundaries leads to the characterization of objects as well as to the understanding of shape properties. The literature on graphs and networks have contributed to the structural characterization of shapes…
Geometric trees are characterized by their tree-structured layout and spatially constrained nodes and edges, which significantly impacts their topological attributes. This inherent hierarchical structure plays a crucial role in domains such…
In the shape analysis approach to computer vision problems, one treats shapes as points in an infinite-dimensional Riemannian manifold, thereby facilitating algorithms for statistical calculations such as geodesic distance between shapes…
Dynamic Time Warping (DTW) and Geometric Edit Distance (GED) are basic similarity measures between curves or general temporal sequences (e.g., time series) that are represented as sequences of points in some metric space $(X,…
In this paper, we prove that the self-dual morphological hierarchical structure computed on a n-D gray-level wellcomposed image u by the algorithm of G{\'e}raud et al. [1] is exactly the mathematical structure defined to be the tree of…
The proper Euclidean geometry is considered to be metric space and described in terms of only metric and finite metric subspaces (sigma-immanent description). Constructing the geometry, one does not use topology and topological properties.…
We introduce a novel representation and optimization framework for discrete geodesics on triangle meshes that reduces artifacts of linear methods on uneven and coarse discretizations. Our method computes squared geodesic distances from…
The graph edit distance is used for comparing graphs in various domains. Due to its high computational complexity it is primarily approximated. Widely-used heuristics search for an optimal assignment of vertices based on the distance…
We use the notion of topological data analysis to compare metrics on data sets. We provide two different motivating examples for this. The first of these is a point cloud data set that has $\mathbb{R}^2$ as its ambient space, and is…
In this paper we study the geometry of graph spaces endowed with a special class of graph edit distances. The focus is on geometrical results useful for statistical pattern recognition. The main result is the Graph Representation Theorem.…
These are the proceedings of the workshop "Math in the Black Forest", which brought together researchers in shape analysis to discuss promising new directions. Shape analysis is an inter-disciplinary area of research with theoretical…
Spatial networks have recently attracted great interest in various fields of research. While the traditional network-theoretic viewpoint is commonly restricted to their topological characteristics (often disregarding existing spatial…
A quasi-metric is a distance function which satisfies the triangle inequality but is not symmetric: it can be thought of as an asymmetric metric. The central result of this thesis, developed in Chapter 3, is that a natural correspondence…
A metric tree ($M$, $d$), also known as $\mathbb{R}$-trees or $T$-theory, is a metric space such that between any two points there is an unique arc and that arc is isometric to an interval in $\mathbb{R}$. In this paper after presenting…
Graph Edit Distance (GED) is defined as the minimum cost transformation of one graph into another and is a widely adopted metric for measuring the dissimilarity between graphs. The major problem of GED is that its computation is NP-hard,…
Continual learning systems operating in fixed-dimensional spaces face a fundamental geometric barrier: the flat manifold problem. When experience is represented as a linear trajectory in Euclidean space, the geodesic distance between…
Understanding the response of an output variable to multi-dimensional inputs lies at the heart of many data exploration endeavours. Topology-based methods, in particular Morse theory and persistent homology, provide a useful framework for…