Related papers: Metric Spaces with Expensive Distances

We consider machine learning in a comparison-based setting where we are given a set of points in a metric space, but we have no access to the actual distances between the points. Instead, we can only ask an oracle whether the distance…

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 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…

Spanners for metric spaces have been extensively studied, both in general metrics and in restricted classes, perhaps most notably in low-dimensional Euclidean spaces -- due to their numerous applications. Euclidean spanners can be viewed as…

We consider the problem of privately answering queries defined on databases which are collections of points belonging to some metric space. We give simple, computationally efficient algorithms for answering distance queries defined over an…

Given a metric space $\mathcal{M}=(X,\delta)$, a weighted graph $G$ over $X$ is a metric $t$-spanner of $\mathcal{M}$ if for every $u,v \in X$, $\delta(u,v)\le d_G(u,v)\le t\cdot \delta(u,v)$, where $d_G$ is the shortest path metric in $G$.…

In this paper we show how the complexity of performing nearest neighbor (NNS) search on a metric space is related to the expansion of the metric space. Given a metric space we look at the graph obtained by connecting every pair of points…

Spatial approximations have been traditionally used in spatial databases to accelerate the processing of complex geometric operations. However, approximations are typically only used in a first filtering step to determine a set of candidate…

Several researchers proposed using non-Euclidean metrics on point sets in Euclidean space for clustering noisy data. Almost always, a distance function is desired that recognizes the closeness of the points in the same cluster, even if the…

Data-sensitive metrics adapt distances locally based the density of data points with the goal of aligning distances and some notion of similarity. In this paper, we give the first exact algorithm for computing a data-sensitive metric called…

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.…

Geometry and topology have generated impacts far beyond their pure mathematical primitive, providing a solid foundation for many applicable tools. Typically, real-world data are represented as vectors, forming a linear subspace for a given…

In this paper, we propose a metric on the space of finite sets of trajectories for assessing multi-target tracking algorithms in a mathematically sound way. The main use of the metric is to compare estimates of trajectories from different…

Analyzing high-dimensional data with manifold learning algorithms often requires searching for the nearest neighbors of all observations. This presents a computational bottleneck in statistical manifold learning when observations of…

We consider the $(1+\epsilon)$-approximate nearest neighbor search problem: given a set $X$ of $n$ points in a $d$-dimensional space, build a data structure that, given any query point $y$, finds a point $x \in X$ whose distance to $y$ is…

The definition of weighted distance measure involves weights. The paper proposes a weighted distance measure without the help of weights. Here, weights are intrinsically added to the measure, and for this, the concept of metric space is…

Learning the embedding space, where semantically similar objects are located close together and dissimilar objects far apart, is a cornerstone of many computer vision applications. Existing approaches usually learn a single metric in the…

Distance metric learning can be viewed as one of the fundamental interests in pattern recognition and machine learning, which plays a pivotal role in the performance of many learning methods. One of the effective methods in learning such a…

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…

$\newcommand{\eps}{\varepsilon}$ In this paper, we consider two important problems defined on finite metric spaces, and provide efficient new algorithms and approximation schemes for these problems on inputs given as graph shortest path…