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We analytically study proximity and distance properties of various kernels and similarity measures on graphs. This helps to understand the mathematical nature of such measures and can potentially be useful for recommending the adoption of…
Edit distance is a measurement of similarity between two sequences such as strings, point sequences, or polygonal curves. Many matching problems from a variety of areas, such as signal analysis, bioinformatics, etc., need to be solved in a…
Proximity graph-based methods have emerged as a leading paradigm for approximate nearest neighbor (ANN) search in the system community. This paper presents fresh insights into the theoretical foundation of these methods. We describe an…
Storing information in DNA molecules is of great interest because of its advantages in longevity, high storage density, and low maintenance cost. A key step in the DNA storage pipeline is to efficiently cluster the retrieved DNA sequences…
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,…
Distance transformation is an image processing technique used for many different applications. Related to a binary image, the general idea is to determine the distance of all background points to the nearest object point (or vice versa). In…
As is well known, classical systems approximate quantum ones -- but how well? We introduce a definition of a "distance" on classical and quantum phase spaces that offers a measure of their separation. Such a distance scale provides a means…
Distances are pervasive in machine learning. They serve as similarity measures, loss functions, and learning targets; it is said that a good distance measure solves a task. When defining distances, the triangle inequality has proven to be a…
Many metric learning tasks, such as triplet learning, nearest neighbor retrieval, and visualization, are treated primarily as embedding tasks where the ultimate metric is some variant of the Euclidean distance (e.g., cosine or Mahalanobis),…
We study the problem of recovering a globally consistent Euclidean embedding of data, given only a local distance graph and propose a method that optimally represents these distances. The method operates solely on a neighborhood graph…
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…
We show new applications of the nearest-neighbor chain algorithm, a technique that originated in agglomerative hierarchical clustering. We apply it to a diverse class of geometric problems: we construct the greedy multi-fragment tour for…
We describe an efficient and scalable spherical graph embedding method. The method uses a generalization of the Euclidean stress function for Multi-Dimensional Scaling adapted to spherical space, where geodesic pairwise distances are…
Measuring similarity between complex objects is a fundamental task in many scientific fields. When objects are represented as graphs, graph similarity/distance measures offer a powerful framework for quantifying structural resemblance.…
The k-nearest neighbors (k-NN) is a basic machine learning (ML) algorithm, and several quantum versions of it, employing different distance metrics, have been presented in the last few years. Although the Euclidean distance is one of the…
Distance plays a fundamental role in measuring similarity between objects. Various visualization techniques and learning tasks in statistics and machine learning such as shape matching, classification, dimension reduction and clustering…
Many methods have been developed for data clustering, such as k-means, expectation maximization and algorithms based on graph theory. In this latter case, graphs are generally constructed by taking into account the Euclidian distance as a…
We study metric data structures for curves in doubling spaces, such as trajectories of moving objects in Euclidean $\mathbb{R}^d$, where the distance between two curves is measured using the discrete Fr\'echet distance. We design data…
Graph comparison plays a major role in many network applications. We often need a similarity metric for comparing networks according to their structural properties. Various network features - such as degree distribution and clustering…
Many statistical and machine learning approaches rely on pairwise distances between data points. The choice of distance metric has a fundamental impact on performance of these procedures, raising questions about how to appropriately…