Related papers: Metrics Based on Average Distance Between Sets
We extend a result, due to Mattila and Sjolin, which says that if the Hausdorff dimension of a compact set $E \subset {\Bbb R}^d$, $d \ge 2$, is greater than $\frac{d+1}{2}$, then the distance set $\Delta(E)=\{|x-y|: x,y \in E \}$ contains…
The metric dimension of a graph is the size of the smallest set of vertices whose distances distinguish all pairs of vertices in the graph. We show that this graph invariant may be calculated by an algorithm whose running time is linear in…
Geometric characteristics of metric spaces that appear in formulas of the Gromov--Hausdorff distances from these spaces to so-called simplexes, i.e., to the metric spaces, all whose non-zero distances are the same are studied. The…
We first consider interval partitions whose complements are Lebesgue-null and introduce a complete metric that induces the same topology as the Hausdorff distance (between complements). This is done using correspondences between intervals.…
Measuring inter-dataset similarity is an important task in machine learning and data mining with various use cases and applications. Existing methods for measuring inter-dataset similarity are computationally expensive, limited, or…
The Hausdorff distance measures how far apart two sets are in a common metric space. By contrast, the Gromov-Hausdorff distance provides a notion of distance between two abstract metric spaces. How do these distances behave for quotients of…
It is shown that given a set of $N$ points in the plane or on the sphere, there is a subset of size $\gtrsim N^{1/3}/\log N$ with all pairwise distances between points distinct.
Measuring visual similarity between two or more instances within a data distribution is a fundamental task in image retrieval. Theoretically, non-metric distances are able to generate a more complex and accurate similarity model than metric…
We present several natural notions of distance between spectral density functions of (discrete-time) random processes. They are motivated by certain filtering problems. First we quantify the degradation of performance of a predictor which…
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…
The notion of task similarity is at the core of various machine learning paradigms, such as domain adaptation and meta-learning. Current methods to quantify it are often heuristic, make strong assumptions on the label sets across the tasks,…
Functions whose composition with every metric is a metric are said to be metric-preserving. In this article, we investigate a variation of the concept of metric-preserving functions where metrics are replaced by ultrametrics.
The concept of depth has proved very important for multivariate and functional data analysis, as it essentially acts as a surrogate for the notion a ranking of observations which is absent in more than one dimension. Motivated by the rapid…
This work builds a unified framework for the study of quadratic form distance measures as they are used in assessing the goodness of fit of models. Many important procedures have this structure, but the theory for these methods is dispersed…
Geometric graphs appear in many real-world data sets, such as road networks, sensor networks, and molecules. We investigate the notion of distance between embedded graphs and present a metric to measure the distance between two geometric…
Distance-based metrics, such as the Hausdorff distance (HD), are widely used to validate segmentation performance in (bio)medical imaging. However, their implementation is complex, and critical differences across open-source tools remain…
A metric space (X,d) is monotone if there is a linear order < on X and a constant c>0 such that d(x,y) < c d(x,z) for all x<y<z in X. Properties of continuous functions with monotone graph (considered as a planar set) are investigated. It…
There are many papers studying properties of point sets in the Euclidean space $\mathbb{E}^m$ or on integer grids $\mathbb{Z}^m$, with pairwise integral or rational distances. In this article we consider the distances or coordinates of the…
Distance correlation is a recent extension of Pearson's correlation, that characterises general statistical independence between Euclidean-space-valued random variables, not only linear relations. This review delves into how and when…
In general, synthesis models provide the mean value of the distribution of possible integrated luminosities, this distribution (and not only its mean value) being the actual description of the integrated luminosity. Therefore, to obtain the…