Related papers: On Locality-Sensitive Orderings and their Applicat…
In [Computer Aided Geometric Design 27 (2010), 212-231] the authors present an algorithm to parametrize approximately $\epsilon$-rational curves, and they show in 2 examples that the Hausdorff distance, w.r.t. to the Euclidean distance,…
The paper introduces a special case of the Euclidean distance matrix completion problem (edmcp) of interest in statistical data analysis where only the minimal spanning tree distances are given and the matrix completion must preserve the…
Designing complex engineered systems requires managing tightly coupled trade-offs between subsystem capabilities and resource requirements. Monotone co-design provides a compositional language for such problems, but its generality does not…
We study $q$-ary codes with distance defined by a partial order of the coordinates of the codewords. Maximum Distance Separable (MDS) codes in the poset metric have been studied in a number of earlier works. We consider codes that are close…
Canonical orderings and their relatives such as st-numberings have been used as a key tool in algorithmic graph theory for the last decades. Recently, a unifying concept behind all these orders has been shown: they can be described by a…
We construct a class of real-valued nonnegative binary functions on a set of jointly distributed random variables, which satisfy the triangle inequality and vanish at identical arguments (pseudo-quasi-metrics). These functions are useful in…
Many geometric optimization problems can be reduced to finding points in space (centers) minimizing an objective function which continuously depends on the distances from the centers to given input points. Examples are $k$-Means, Geometric…
We study the problem of finding a mapping $f$ from a set of points into the real line, under ordinal triple constraints. An ordinal constraint for a triple of points $(u,v,w)$ asserts that $|f(u)-f(v)|<|f(u)-f(w)|$. We present an…
The distances between flats of a Poisson $k$-flat process in the $d$-dimensional Euclidean space with $k<d/2$ are discussed. Continuing an approach originally due to Rolf Schneider, the number of pairs of flats having distance less than a…
Let $X$ be an isotropic random vector in $R^d$ that satisfies that for every $v \in S^{d-1}$, $\|<X,v>\|_{L_q} \leq L \|<X,v>\|_{L_p}$ for some $q \geq 2p$. We show that for $0<\varepsilon<1$, a set of $N = c(p,q,\varepsilon) d$ random…
LP-type problems such as the Minimum Enclosing Ball (MEB), Linear Support Vector Machine (SVM), Linear Programming (LP), and Semidefinite Programming (SDP) are fundamental combinatorial optimization problems, with many important…
When we represent a network of sensors in Euclidean space by a graph, there are two distances between any two nodes that we may consider. One of them is the Euclidean distance. The other is the distance between the two nodes in the graph,…
The Wasserstein distance received a lot of attention recently in the community of machine learning, especially for its principled way of comparing distributions. It has found numerous applications in several hard problems, such as domain…
We consider the popular and classical method of alternating projections for finding a point in the intersection of two closed sets. By situating the algorithm in a metric space, equipped only with well-behaved geodesics and angles (in the…
Orders with low crossing number, introduced by Welzl, are a fundamental tool in range searching and computational geometry. Recently, they have found important applications in structural graph theory: set systems with linear shatter…
Random geometric graphs result from taking $n$ uniformly distributed points in the unit cube, $[0,1]^d$, and connecting two points if their Euclidean distance is at most $r$, for some prescribed $r$. We show that monotone properties for…
Consider that the coordinates of $N$ points are randomly generated along the edges of a $d$-dimensional hypercube (random point problem). The probability that an arbitrary point is the $m$th nearest neighbor to its own $n$th nearest…
We consider logics derived from Euclidean spaces $\mathbb{R}^n$. Each Euclidean space carries relations consisting of those pairs that are, respectively, distance more than 1 apart, distance less than 1 apart, and distance 1 apart. Each…
Image ranking is to rank images based on some known ranked images. In this paper, we propose an improved linear ordinal distance metric learning approach based on the linear distance metric learning model. By decomposing the distance metric…
We present an algorithm for efficiently approximating of qubit unitaries over gate sets derived from totally definite quaternion algebras. It achieves $\varepsilon$-approximations using circuits of length $O(\log(1/\varepsilon))$, which is…