Related papers: On distances and metrics in discrete ordered sets

In 2017 M. Bessenyei and Z. P\'ales introduced a definition of a triangle function which generates a concept of a generalized triangle inequality in semimetric spaces. Inspired by this concept we discuss already known inequalities in metric…

Metric magnitude is a measure of the "size" of point clouds with many desirable geometric properties. It has been adapted to various mathematical contexts and recent work suggests that it can enhance machine learning and optimization…

We consider the problem of optimizing the product of the distances from a given point in a triangle to each vertex. There are two possible cases in general. For isosceles triangles, we explicitly show exactly when both cases occur.

We determine the maximum distance between any two of the center, centroid, and subtree core among trees with a given order. Corresponding results are obtained for trees with given maximum degree and also for trees with given diameter. The…

In this paper, we study a classical construction of lattices from number fields and obtain a series of new results about their minimum distance and other characteristics by introducing a new measure of algebraic numbers. In particular, we…

We outline a proof of an analogue of Khintchine's Theorem in R^2, where the ordinary height is replaced by a distance function satisfying an irrationality condition as well as certain decay and symmetry conditions.

Looking at some monoids and (semi)rings (natural numbers, integers and p-adic integers), and more generally, residually finite algebras (in a strong sense), we prove the equivalence of two ways for a function on such an algebra to behave…

Given a set of points in the Euclidean space $\mathbb{R}^\ell$ with $\ell>1$, the pairwise distances between the points are determined by their spatial location and the metric $d$ that we endow $\mathbb{R}^\ell$ with. Hence, the distance…

Probabilistic submeasures generalizing the classical (numerical) submeasures are introduced and discussed in connection with some classes of aggregation functions. A special attention is paid to triangular norm-based probabilistic…

Embedding data into vector spaces is a very popular strategy of pattern recognition methods. When distances between embeddings are quantized, performance metrics become ambiguous. In this paper, we present an analysis of the ambiguity…

Optimal transport distances have found many applications in machine learning for their capacity to compare non-parametric probability distributions. Yet their algorithmic complexity generally prevents their direct use on large scale…

Distance metric learning can be viewed as one of the fundamental interests in pattern recognition and machine learning, which plays a pivotal role in the performance of many learning methods. One of the effective methods in learning such a…

Let $X$ be an ordered vector space. The net $\{x_\alpha\}\subseteq X$ is semi unbounded order convergent to $x$ (in symbol $x_\alpha\xrightarrow{suo}x$), if there is a net $\{y_\beta\}$, possibly over a different index set, such that…

A distance-based inconsistency indicator, defined by the third author for the consistency-driven pairwise comparisons method, is extended to the incomplete case. The corresponding optimization problem is transformed into an equivalent…

We initiate a systematic study of lattices of thick subcategories for arbitrary essentially small triangulated categories. To this end we give several examples illustrating the various properties these lattices may, or may not, have and…

Chain distance between points in a metric space is defined as the infimum of epsilon such that there is an epsilon-chain connecting these points. We call a mapping of a metric compact into the real line a chain development if it preserves…

Metric learning makes it plausible to learn distances for complex distributions of data from labeled data. However, to date, most metric learning methods are based on a single Mahalanobis metric, which cannot handle heterogeneous data well.…

We present a quasi-integrable two-dimensional lattice equation: i.e., a partial difference equation which satisfies a criterion of integrability, singularity confinement, although it has a chaotic aspect in the sense that the degrees of its…

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…

This paper is concerned with the study of scalability in nonlinear heterogeneous networks affected by communication delays and disturbances. After formalizing the notion of scalability, we give two sufficient conditions to assess this…