Related papers: Subtree Distances, Tight Spans and Diversities
Many variants of Optimal Transport (OT) have been developed to address its heavy computation. Among them, notably, Sliced Wasserstein (SW) is widely used for application domains by projecting the OT problem onto one-dimensional lines, and…
Metric learning has the aim to improve classification accuracy by learning a distance measure which brings data points from the same class closer together and pushes data points from different classes further apart. Recent research has…
The metric dimension has been introduced independently by Harary, Melter and Slater in 1975 to identify vertices of a graph G using its distances to a subset of vertices of G. A resolving set X of a graph G is a subset of vertices such…
The metric dimension of a graph is the smallest number of nodes required to identify all other nodes based on shortest path distances uniquely. Applications of metric dimension include discovering the source of a spread in a network,…
This is a write-up of a talk given at the CATMI meeting in Bergen in July 2023, and is an introduction to a category-theoretic perspective on metric spaces. A metric space is a set of points such that between each pair of points there is a…
Euclidean embeddings of data are fundamentally limited in their ability to capture latent semantic structures, which need not conform to Euclidean spatial assumptions. Here we consider an alternative, which embeds data as discrete…
Let $\mathfrak{M}$ be a class of metric spaces. A metric space $Y$ is minimal $\mathfrak{M}$-universal if every $X\in\mathfrak{M}$ can be isometrically embedded in $Y$ but there are no proper subsets of $Y$ satisfying this property. We find…
We propose a consistent approach to the statistics of the shortest paths in random graphs with a given degree distribution. This approach goes further than a usual tree ansatz and rigorously accounts for loops in a network. We calculate the…
In this work, a new way to represent Japanese animation (anime) is presented. We applied a minimum spanning tree to show the relation between anime. The distance between anime is calculated through three similarity measurements, namely…
Among many topological indices of trees the sum of distances $\sigma(T)$ and the number of subtrees $F(T)$ have been a long standing pair of graph invariants that are well known for their negative correlation. That is, among various given…
Diversities are like metric spaces, except that every finite subset, instead of just every pair of points, is assigned a value. Just as there is a theory of minimal distortion embeddings of finite metric spaces into $L_1$, there is a…
Ultametrics are an important class of distances used in applications such as phylogenetics, clustering and classification theory. Ultrametrics are essentially distances that can be represented by an edge-weighted rooted tree so that all of…
Modeling place functions from a computational perspective is a prevalent research topic. Trajectory embedding, as a neural-network-backed dimension reduction technology, allows the possibility to put places with similar social functions at…
It is shown that a minimum weight spanning tree of a finite ultrametric space can be always found in the form of path. As a canonical representing tree such path uniquely defines the whole space and, moreover, it has much more simple…
The threshold-$k$ metric dimension ($\mathrm{Tmd}_k$) of a graph is the minimum number of sensors -- a subset of the vertex set -- needed to uniquely identify any vertex in the graph, solely based on its distances from the sensors, when the…
Deep metric learning aims to construct an embedding space where samples of the same class are close to each other, while samples of different classes are far away from each other. Most existing deep metric learning methods attempt to…
Merge trees are fundamental structures in topological data analysis. Interleaving distance is a widely accepted metric for comparing merge trees, with applications in visualization and scientific computing. While a greedy algorithm exists…
The smallest hyperconvex metric space containing a given metric space X is called the tight span of X. It is known that tight spans have many nice geometric and topological properties, and they are gradually becoming a target of research of…
The Fr\'echet distance is a popular distance measure for curves which naturally lends itself to fundamental computational tasks, such as clustering, nearest-neighbor searching, and spherical range searching in the corresponding metric…
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…