Related papers: Generalization of distance to higher dimensional o…
There are many methods developed to approximate a cloud of vectors embedded in high-dimensional space by simpler objects: starting from principal points and linear manifolds to self-organizing maps, neural gas, elastic maps, various types…
The distance standard deviation, which arises in distance correlation analysis of multivariate data, is studied as a measure of spread. The asymptotic distribution of the empirical distance standard deviation is derived under the assumption…
One of the hallmarks of quantum theory is the realization that distinct measurements cannot in general be performed simultaneously, in stark contrast to classical physics. In this context the notions of coexistence and joint measurability…
A method is proposed that allows one to infer the sum of the values of an observable taken during contacts with a pointer state. Hereby the state of the pointer is updated while contacted with the system and remains unchanged between…
The depinning of an elastic line interacting with a quenched disorder is studied for long range interactions, applicable to crack propagation or wetting. An ultrametric distance is introduced instead of the Euclidean distance, allowing for…
Hanika, Schneider, and Stumme introduced geometric data set as a generalization of metric measure space for the computation of the observable diameter, and extended the observable distance between metric measure spaces to that between…
What is the distance between two points in spacetime? This is a basic geometric question, which so far has no single, definitive answer. Unlike their Riemannian cousins, Lorentzian manifolds are not known to carry a canonical distance…
The concepts of symmetry and its breakdown are investigated in two different terms according to whether the resulting asymmetry is universal or only obtained for a special configuration: we shall illustrate this by considering in the first…
We introduce the so--called doubling metric on the collection of non--empty bounded open subsets of a metric space. Given a subset $U$ of a metric space $X$, the predecessor $U_{*}$ of $U$ is defined by doubling the radii of all open balls…
In this short technical report, we define on the sample space R^D a distance between data points which depends on their correlation. We also derive an expression for the center of mass of a set of points with respect to this distance.
Many classical geometric inequalities on functionals of convex bodies depend on the dimension of the ambient space. We show that this dimension dependence may often be replaced (totally or partially) by different symmetry measures of the…
A novel measure, quantumness of correlations is introduced here for bipartite states, by incorporating the required measurement scheme crucial in defining any such quantity. Quantumness coincides with the previously proposed measures in…
The generalized divided differences are introduced. They are applied to investigate some properties characterizing generalized higher-order convexity. Among others some support-type property is proved.
We discuss why regular observables can not be proper entanglement measures, and how observables in a generalized setting can be used to make an entanglement monotone a directly observable quantity for the case of pure states. For the case…
We consider mappings satisfying an upper bound for the distortion of families of curves. We establish lower bounds for the distortion of distances under such mappings. As applications, we obtain theorems on the discreteness of the limit…
We present a generalization of the Holevo theorem by means of distances used in the definition of distinguishability of states, showing that each one leads to an alternative Holevo theorem. This result involves two quantities: the…
Understanding distance metrics in high-dimensional spaces is crucial for various fields such as data analysis, machine learning, and optimization. The Manhattan distance, a fundamental metric in multi-dimensional settings, measures the…
Similarity between objects is multi-faceted and it can be easier for human annotators to measure it when the focus is on a specific aspect. We consider the problem of mapping objects into view-specific embeddings where the distance between…
We study the isoperimetric problem in product spaces equipped with the uniform distance. Our main result is a characterization of isoperimetric inequalities which, when satisfied on a space, are still valid for the product spaces, up a to a…
We introduce a generalization for bounded geometry that we call bounded scale measure. We show that bounded scale measure is a coarse invariant unlike bounded geometry. We then show equivalent definitions for spaces with bounded scale…