Related papers: The Distance Function on a Computable Graph
An ordinal-valued metric taking its values in the set of all countable ordinals can be assigned to a metrizable set of nodes in a transfinite graph. Then, a variety of results concerning nodal eccentricities, radii, diameters, centers,…
Any function can be constructed using a hierarchy of simpler functions through compositions. Such a hierarchy can be characterized by a binary rooted tree. Each node of this tree is associated with a function which takes as inputs two…
Given a continuous function $f:[a,b]\to\mathbb{R}$ such that $f(a)=f(b)$, we investigate the set of distances $|x-y|$ where $f(x)=f(y)$. In particular, we show that the only distances this set must contain are ones which evenly divide…
We consider isomorphism of controllable graphs and cospectrality of distance-regularized graphs (which are known to be distance-regular or distance-biregular) in relation to logical definability. While most characterizations of these…
The \emph{distance-number} of a graph $G$ is the minimum number of distinct edge-lengths over all straight-line drawings of $G$ in the plane. This definition generalises many well-known concepts in combinatorial geometry. We consider the…
The notions of distance and similarity play a key role in many machine learning approaches, and artificial intelligence (AI) in general, since they can serve as an organizing principle by which individuals classify objects, form concepts…
The paper deals with the problem of reconstructing the topological structure of a network of dynamical systems. A distance function is defined in order to evaluate the "closeness" of two processes and a few useful mathematical properties…
We are interested in the general question: to what extent are the spectral properties of a graph connected to the distance function? Our motivation is a concrete example of this question that is due to Steinerberger. We provide some…
We prove the computability of a version of Whitney Extension, when the input is suitably represented. More specifically, if $F \subseteq \mathbb{R}^n$ is a closed set represented so that the distance function $x \mapsto d(x,F)$ can be…
A distance-squared function is one of the most significant functions in the application of singularity theory to differential geometry. In this paper, we define naturally extended mappings of distance-squared functions, wherein each…
Computing the distance function to some surface or line is a problem that occurs very frequently. There are several ways of computing a relevant approximation of this function, using for example technique originating from the approximation…
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…
We derive computationally tractable methods to select a small subset of experiment settings from a large pool of given design points. The primary focus is on linear regression models, while the technique extends to generalized linear models…
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…
A geometric graph is a combinatorial graph, endowed with a geometry that is inherited from its embedding in a Euclidean space. Formulation of a meaningful measure of (dis-)similarity in both the combinatorial and geometric structures of two…
Many functions have been recently defined to assess the similarity among networks as tools for quantitative comparison. They stem from very different frameworks - and they are tuned for dealing with different situations. Here we show an…
We study the complexity of approximations to the normalized information distance. We introduce a hierarchy of computable approximations by considering the number of oscillations. This is a function version of the difference hierarchy for…
A well-defined distance on the parameter space is key to evaluating estimators, ensuring consistency, and building confidence sets. While there are typically standard distances to adopt in a continuous space, this is not the case for…
Submodular functions are a fundamental object of study in combinatorial optimization, economics, machine learning, etc. and exhibit a rich combinatorial structure. Many subclasses of submodular functions have also been well studied and…
We give an overview of different approaches to measuring the similarity of, or the distance between, two graphs, highlighting connections between these approaches. We also discuss the complexity of computing the distances.