Related papers: Linear and Sublinear Diversities
Matrix convexity generalizes convexity to the dimension free setting and has connections to many mathematical and applied pursuits including operator theory, quantum information, noncommutative optimization, and linear control systems. In…
We describe all metric spaces that have sufficently many affine functions. As an application we obtain a metric characterization of linear-convex subsets of Banach spaces.
Let $n,p,r$ be positive integers with $n \geq p\geq r$. A rank-$\overline{r}$ subset of $n$ by $p$ matrices (with entries in a field) is a subset in which every matrix has rank less than or equal to $r$. A classical theorem of Flanders…
We study a long-recognised but under-appreciated symmetry called "dynamical similarity" and illustrate its relevance to many important conceptual problems in fundamental physics. Dynamical similarities are general transformations of a…
The Solow--Polasky diversity indicator (or magnitude) is a classical measure of diversity based on pairwise distances. It has applications in ecology, conservation planning, and, more recently, in algorithmic subset selection and diversity…
Let $r$ and $n$ be positive integers such that $r<n$, and $\mathbb{K}$ be an arbitrary field. We determine the maximal dimension for an affine subspace of $n$ by $n$ symmetric (or alternating) matrices with entries in $\mathbb{K}$ and with…
This paper concerns the \textbf{abstract geometry of numbers}: namely the pursuit of certain aspects of geometry of numbers over a suitable class of normed domains. (The standard geometry of numbers is then viewed as geometry of numbers…
We examine the total mixed scalar curvature of a fixed distribution as a functional of a pseudo-Riemannian metric. We develop variational formulas for quantities of extrinsic geometry of the distribution to find the critical points of this…
Diversity can be broadly defined as the presence of meaningful variation across elements, which can be viewed from multiple perspectives, including statistical variation and geometric structural richness in the dataset. Existing diversity…
Consider a convex function that is invariant under an group of transformations. If it has a minimizer, does it also have an invariant minimizer? Variants of this problem appear in nonparametric statistics and in a number of adjacent fields.…
A complete classification is obtained of continuous, translation invariant, Minkowski valuations on an m-dimensional complex vector space which are covariant under the complex special linear group.
The paper introduces the class of O-metric spaces, a novel generalization of metric-type spaces, classifying almost all possible metric types into upward and downward O-metrics. We list some topologies arising from O-metrics and discuss…
Multidimensional scaling (MDS) is a popular technique for mapping a finite metric space into a low-dimensional Euclidean space in a way that best preserves pairwise distances. We study a notion of MDS on infinite metric measure spaces,…
In the first part of the paper, we define an approximated Brunn-Minkowski inequality which generalizes the classical one for length spaces. Our new definition based only on distance properties allows us also to deal with discrete spaces.…
The paper deals with a comprehensive theory of mappings, whose local behavior can be described by means of linear subspaces, contained in the graphs of two (primal and dual) generalized derivatives. This class of mappings includes the…
Pairs of metrics in a three-dimensional linear vector space are considered, one of which is a Minkowski type metric with the signature (+,-,-). Such metric pairs are classified and canonical presentations for them in each class are…
With the emergence of deep learning, metric learning has gained significant popularity in numerous machine learning tasks dealing with complex and large-scale datasets, such as information retrieval, object recognition and recommendation…
Mixture distributions are extensively used as a modeling tool in diverse areas from machine learning to communications engineering to physics, and obtaining bounds on the entropy of probability distributions is of fundamental importance in…
We present several applications of matrix-theoretic inequalities to the magnitude of metric spaces. We first resolve an open problem by showing that the magnitude of any finite metric space of negative type is less than or equal to its…
Linear topological spaces with partial ordering (linear kinematics) are studied. They are defined by a set of 8 axioms implying that topology, linear structure and ordering are compatible with each other. Most of the results are valid for…