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Metric embeddings are central to metric theory and its applications. Here we consider embeddings of a different sort: maps from a set to subsets of a metric space so that distances between points are approximated by minimal distances…
We study the problem of finding the $k$ most similar trajectories to a given query trajectory. Our work is inspired by the work of Grossi et al. [6] that considers trajectories as walks in a graph. Each visited vertex is accompanied by a…
Fitting distances to tree metrics and ultrametrics are two widely used methods in hierarchical clustering, primarily explored within the context of numerical taxonomy. Given a positive distance function…
We study the Approximate Nearest Neighbor problem for metric spaces where the query points are constrained to lie on a subspace of low doubling dimension, while the data is high-dimensional. We show that this problem can be solved…
We initiate the rigorous study of classification in quasi-metric spaces. These are point sets endowed with a distance function that is non-negative and also satisfies the triangle inequality, but is asymmetric. We develop and refine a…
Let $\theta\in\mathbb{R}^d$. We associate three objects to each approximation $(p,q)\in \mathbb{Z}^d\times \mathbb{N}$ of $\theta$: the projection of the lattice $\mathbb{Z}^{d+1}$ to the hyperplane of the first $d$ coordinates along the…
We survey a new area of parameter-free similarity distance measures useful in data-mining, pattern recognition, learning and automatic semantics extraction. Given a family of distances on a set of objects, a distance is universal up to 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…
Organizing large-scale resources in a multidimensional semantic space is an approach to efficiently managing and querying resources from different semantic dimensions. To support advanced applications, this paper proposes a resource space…
Data depth is a concept in multivariate statistics that measures the centrality of a point in a given data cloud in $\IR^d$. If the depth of a point can be represented as the minimum of the depths with respect to all one-dimensional…
We consider the following $q$-analog of the basic combinatorial search problem: let $q$ be a prime power and $\GF(q)$ the finite field of $q$ elements. Let $V$ denote an $n$-dimensional vector space over $\GF(q)$ and let $\mathbf{v}$ be an…
The modern image search system requires semantic understanding of image, and a key yet under-addressed problem is to learn a good metric for measuring the similarity between images. While deep metric learning has yielded impressive…
We outline a new approach for solving optimization problems which enforce triangle inequalities on output variables. We refer to this as metric-constrained optimization, and give several examples where problems of this form arise in machine…
A classical vector retrieval problem typically considers a \emph{single} query embedding vector as input and retrieves the most similar embedding vectors from a vector database. However, complex reasoning and retrieval tasks frequently…
We consider the following generalization of binary search in sorted arrays to tree domains. In each step of the search, an algorithm is querying a vertex $q$, and as a reply, it receives an answer, which either states that $q$ is the…
In this paper we propose and study a new complexity model for approximation algorithms. The main motivation are practical problems over large data sets that need to be solved many times for different scenarios, e.g., many multicast trees…
We present a scalable approach for range and $k$ nearest neighbor queries under computationally expensive metrics, like the continuous Fr\'echet distance on trajectory data. Based on clustering for metric indexes, we obtain a dynamic tree…
When reasoning about tasks that involve large amounts of data, a common approach is to represent data items as objects in the Hamming space where operations can be done efficiently and effectively. Object similarity can then be computed by…
We prove that if X is a complete geodesic metric space with uniformly generated first homology group and $f: X\to R$ is metrically proper on the connected components and bornologous, then X is quasi-isometric to a tree. Using this and…
Many datasets exhibit a well-defined structure that can be exploited to design faster search tools, but it is not always clear when such acceleration is possible. Here, we introduce a framework for similarity search based on characterizing…