Related papers: Metrics for more than two points at once
In this work, we define the notion of unimodular random measured metric spaces as a common generalization of various other notions. This includes the discrete cases like unimodular graphs and stationary point processes, as well as the…
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…
Many organizations describe their processes as consensus-driven, but there is no consensus on the definition of consensus. Qualitative definitions of consensus prioritize social phenomena like "unity" that are not necessarily measurable.…
In this paper we show that in systems where the probability distribution of the the overlap is non trivial in the infinity volume limit, the property of ultrametricity can be proved in general starting from two very simple and natural…
Probability metrics constitute an important tool in probability theory and statistics \cite{DKS91}, \cite{R91}, \cite{Z83} as they are specific metrics on spaces of random variables which, by satisfying an extra condition, concord well with…
Topic models extract representative word sets - called topics - from word counts in documents without requiring any semantic annotations. Topics are not guaranteed to be well interpretable, therefore, coherence measures have been proposed…
A metric space is indivisible if for any partition of it into finitely many pieces one piece contains an isometric copy of the whole space. Continuing our investigation of indivisible metric spaces, we show that a countable ultrametric…
We prove a central limit theorem for a sequence of random variables whose means are ambiguous and vary in an unstructured way. Their joint distribution is described by a set of measures. The limit is (not the normal distribution and is)…
In this paper we present with algebraic trees a novel notion of (continuum) trees which generalizes countable graph-theoretic trees to (potentially) uncountable structures. For that purpose we focus on the tree structure given by the branch…
In this paper we introduce the concept of the rectangular metric like spaces, along with its topology and we prove some fixed point theorems under different contraction principles. We introduce the concept of modified metric-like space as…
We define a measure of redundant information based on projections in the space of probability distributions. Redundant information between random variables is information that is shared between those variables. But in contrast to mutual…
Although copulas are used and defined for various infinite-dimensional objects (e.g. Gaussian processes and Markov processes), there is no prevalent notion of a copula that unifies these concepts. We propose a unified approach and define…
We study here the topology of information on the space of probability measures over Polish spaces that was defined in [1]. We show that under this topology, a convergent sequence of probability measures satisfying a conditional independence…
In this paper we propose numerical measures for evaluating the aesthetic interest of simple patterns. The patterns consist of elements (symbols, pixels, etc.) in regular square arrays. The measures depend on two characteristics of the…
We study fractal measures on Euclidean space through the dynamics of "zooming in" on typical points. The resulting family of measures (the "scenery"), can be interpreted as an orbit in an appropriate dynamical system which often…
Probability distributions can be read as simple expressions of information. Each continuous probability distribution describes how information changes with magnitude. Once one learns to read a probability distribution as a measurement scale…
Uniform measures are the functionals on the space of bounded uniformly continuous functions that are continuous on every bounded uniformly equicontinuous set. This paper describes the role of uniform measures in the study of convolution on…
Mechanistic interpretability aims to break models into meaningful parts; verifying that two such parts implement the same computation is a prerequisite. Existing similarity measures evaluate either empirical behaviour, leaving them blind to…
Porosity and dimension are two useful, but different, concepts that quantify the size of fractal sets and measures. An active area of research concerns understanding the relationship between these two concepts. In this article we will…
Given a compact pseudo-metric space, we associate to it upper and lower dimensions, depending only on the metric. Then we construct a doubling metric for which the measure of a dillated ball is closely related to these dimensions.