Related papers: Mutual Dimension
Euclidean preferences are a widely studied preference model, in which decision makers and alternatives are embedded in d-dimensional Euclidean space. Decision makers prefer those alternatives closer to them. This model, also known as…
The design of a metric between probability distributions is a longstanding problem motivated by numerous applications in Machine Learning. Focusing on continuous probability distributions on the Euclidean space $\mathbb{R}^d$, we introduce…
We derive a well-defined renormalized version of mutual information that allows to estimate the dependence between continuous random variables in the important case when one is deterministically dependent on the other. This is the situation…
In the recent paper [J. Funct. Anal. {\bf 259} (2010)], Naor and Tao introduce a new class of measures with a so-called micro-doubling property and present, via martingale theory and probability methods, a localization theorem for the…
We prove several inequalities estimating the distance between volumes of two bodies in terms of the maximal or minimal difference between areas of sections or projections of these bodies. We also provide extensions in which volume is…
We address pairs $(X, Y)$ of metric spaces with the following property: for every mapping $f: X \to Y$ the existence of points $x, y \in X$ with $d(f(x),f(y)) > d(x,y)$ implies the existence of $\widetilde{x}, \widetilde{y}\in X$ for which…
Motivated by the manifold hypothesis, which states that data with a high extrinsic dimension may yet have a low intrinsic dimension, we develop refined statistical bounds for entropic optimal transport that are sensitive to the intrinsic…
The evolution of human intelligence led to the huge amount of data in the information space. Accessing and processing this data helps in finding solutions to applied problems based on finite-dimensional models. We argue, that formally, such…
Estimating the dimensionality of the latent representation needed for prediction -- the task-relevant dimension -- is a difficult, largely unsolved problem with broad scientific applications. We cast it as an Information Bottleneck…
In algorithms for finite metric spaces, it is common to assume that the distance between two points can be computed in constant time, and complexity bounds are expressed only in terms of the number of points of the metric space. We…
The development of algorithmic fractal dimensions in this century has had many fruitful interactions with geometric measure theory, especially fractal geometry in Euclidean spaces. We survey these developments, with emphasis on connections…
In the framework of Clifford analysis, a chain of harmonic and monogenic potentials in the upper half of (m+1)-dimensional Euclidean space was recently constructed, including a higher dimensional analogue of the logarithmic function in the…
We study the action of translation on the spaces of uniformly bounded continuous functions on the real line which are uniformly band-limited in a compact interval. We prove that two intervals themselves will decide if two spaces are…
Multivariate pattern analyses approaches in neuroimaging are fundamentally concerned with investigating the quantity and type of information processed by various regions of the human brain; typically, estimates of classification accuracy…
Mutual information (MI) is a fundamental measure of statistical dependence, with a myriad of applications to information theory, statistics, and machine learning. While it possesses many desirable structural properties, the estimation of…
Dimension theory lies at the heart of fractal geometry and concerns the rigorous quantification of how large a subset of a metric space is. There are many notions of dimension to consider, and part of the richness of the subject is in…
We introduce a continuum of dimensions which are `intermediate' between the familiar Hausdorff and box dimensions. This is done by restricting the families of allowable covers in the definition of Hausdorff dimension by insisting that $|U|…
We consider the notion of dimension in four categories: the category of (unbounded) separable metric spaces and (metrically proper) Lipschitz maps, and the category of (unbounded) separable metric spaces and (metrically proper) uniform…
In the framework of Clifford analysis, a chain of harmonic and monogenic potentials in the upper half of Euclidean space R^{m+1} was constructed recently, including a higher dimensional analogue of the logarithmic function in the complex…
Human similarity judgments are inconsistent with Euclidean, Hamming, Mahalanobis, and the majority of measures used in the extensive literatures on similarity and dissimilarity. From intrinsic properties of brain circuitry, we derive…