Related papers: Approximating the Convex Hull via Metric Space Mag…
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…
Magnitude homology is an invariant of enriched categories which generalizes ordinary categorical homology -- the homology of the classifying space of a small category. The classifying space can also be generalized in a different direction:…
We give the first differentially private algorithms that estimate a variety of geometric features of points in the Euclidean space, such as diameter, width, volume of convex hull, min-bounding box, min-enclosing ball etc. Our work relies…
Magnitude is an isometric invariant of metric spaces introduced by Leinster. Although magnitude is nowhere continuous on the Gromov-Hausdorff space of finite metric spaces, continuity results are possible if we restrict the ambient space.…
The convex hull of a set of points, $C$, serves to expose extremal properties of $C$ and can help identify elements in $C$ of high interest. For many problems, particularly in the presence of noise, the true vertex set (and facets) may be…
We give sufficient conditions for a finite metric space to be determined by the magnitude function. In particular, a generic finite metric space such that the distances between the points are rationally independent is determined by the…
In ancient Greek mathematics, magnitudes such as lengths were strictly distinguished from numbers. In modern quantity calculus, a distinction is made between quantities and scalars that serve as measures of quantities. It can be argued that…
In this paper, we present algorithms for computing approximate hulls and centerpoints for collections of matrices in positive definite space. There are many applications where the data under consideration, rather than being points in a…
Finite metric spaces arise in many different contexts. Enormous bodies of data, scientific, commercial and others can often be viewed as large metric spaces. It turns out that the metric of graphs reveals a lot of interesting information.…
A new method of metric space investigation, based on classification of its finite subspaces, is suggested. It admits to derive information on metric space properties which is encoded in metric. The method describes geometry in terms of only…
Magnitude is a real-valued invariant of metric spaces which, in the finite setting, can be understood as recording the 'effective number of points' in a space as the scale of the metric varies. Motivated by applications in topological data…
A new intrinsic volume metric is introduced for the class of convex bodies in $\mathbb{R}^n$. As an application, an inequality is proved for the asymptotic best approximation of the Euclidean unit ball by arbitrarily positioned polytopes…
The concept of boundary plays an important role in several branches of general relativity, e.g., the variational principle for the Einstein equations, the event horizon and the apparent horizon of black holes, the formation of trapped…
There is a general notion of the magnitude of an enriched category, defined subject to hypotheses. In topological and geometric contexts, magnitude is already known to be closely related to classical invariants such as Euler characteristic…
Metric spaces are a fundamental component of mathematics and have a paramount importance as a framework for measuring distance. They can be found in many different branches of mathematics, such as analysis and topology. This paper offers an…
We develop tools for explicitly constructing categories enriched over generating data and that compose via ordinary scalar and matrix arithmetic arithmetic operations. We characterize meaningful size maps, weightings, and magnitude that…
We study the geometric significance of Leinster's notion of magnitude for a smooth manifold with boundary of arbitrary dimension, motivated by open questions for the unit disk in $\mathbb{R}^2$. For a large class of distance functions,…
Recently, motivated by the rapid increase of the data size in various applications, Monemizadeh [APPROX'23] and Driemel, Monemizadeh, Oh, Staals, and Woodruff [SoCG'25] studied geometric problems in the setting where the only access to the…
In this paper we introduce the persistent magnitude, a new numerical invariant of (sufficiently nice) graded persistence modules. It is a weighted and signed count of the bars of the persistence module, in which a bar of the form $[a,b)$ in…
A new approach to the local and global explanation is proposed. It is based on selecting a convex hull constructed for the finite number of points around an explained instance. The convex hull allows us to consider a dual representation of…