A classical result in metric geometry asserts that any n-point metric admits a linear-size spanner of dilation O(log n) [PS89]. More generally, for any c > 1, any metric space admits a spanner of size O(n), and dilation at most c. This bound is tight assuming the well-known girth conjecture of Erdős [Erd63]. We show that for a metric induced by a set of n points in high-dimensional Euclidean sp...

In this work, we define a new metric of the distance and depth of penetration between two convex polyhedral bodies. The metric is computed by means of a linear program with three variables and m+n constraints, where m and n are the number of facets of the two polyhedral bodies. As a result, this metric can be computed with O(n+m) algorithmic complexity, superior to the best algorithms known for...

Existing deep embedding methods in vision tasks are capable of learning a compact Euclidean space from images, where Euclidean distances correspond to a similarity metric. To make learning more effective and efficient, hard sample mining is usually employed, with samples identified through computing the Euclidean feature distance. However, the global Euclidean distance cannot faithfully charact...

Teramoto et al. [TAKD06] defined a new measure called the gap ratio that measures the uniformity of a finite point set sampled from S, a bounded subset of R. We generalize this definition of measure over all metric spaces by appealing to covering and packing radius. The definition of gap ratio needs only a metric unlike discrepancy, a widely used uniformity measure, that depends on the notion o...

We prove a simple, explicit formula for the mass of any asymptotically locally Euclidean (ALE) Kähler manifold, assuming only the sort of weak fall-off conditions required for the mass to actually be well-defined. For ALE scalar-flat Kähler manifolds, the mass turns out to be a topological invariant, depending only on the underlying smooth manifold, the first Chern class of the complex structur...

We construct continuous families of scattering manifolds with the same scattering phase. The manifolds are compactly supported metric perturbations of Euclidean R for n ≥ 8. The metric perturbation may have arbitrarily small support.

We show that on Hilbert scheme of n points on C, the hyperkähler metric construsted by H. Nakajima via hyperkähler reduction is the Quasi-Asymptotically Locally Euclidean (QALE in short) metric constructed by D. Joyce.

Deformable models like snakes are a classical tool for image segmentation. Highly deformable models extend them with the ability to handle dynamic topological changes, and therefore to extract arbitrary complex shapes. However, the resolution of these models largely depends on the resolution of the image. As a consequence, their time and memory complexity increases at least as fast as the size ...