We consider a simple graph-based metric on points in Euclidean space known as the edge-squared metric. This metric is defined by squaring the Euclidean distance between points, and taking the shortest paths on the resulting graph. This metrichas been studied before in wireless networks and machine learning, and has the density sensitive property: distances between two points in the same cluster are short, even if their Euclidean distance is long. This property is desirable in machine learning . In this paper, we show that this metricis equal to previously studied geodesic-based metric defined on points in Euclidean space. Previous best work showed that these metrics are 3-approximations of each other. It was not known or suspected that these metrics were equal. We give fast algorithms to compute sparse spanners of this distance, a problem that can be seen as a generalization of both the Euclidean spanner and the Euclidean MST problem. Spanners of the edge-squared metric are faster to compute and sparser than the best known Euclidean spanners in a variety of settings .
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 the nearest neighbor metric. In fact, we prove the surprising result that a previously published 3-approximation is an exact algorithm. The nearest neighbor metric can be viewed as a special case of a density-based distance used in machine learning , or it can be seen as an example of a manifold metric. Previous computational research on such metrics despaired of computing exact distances on account of the apparent difficulty of minimizing over all continuous paths between a pair of points. We leverage the exact computation of the nearest neighbor metric to compute sparse spanners and persistent homology. We also explore the behavior of the metric built from point sets drawn from an underlying distribution and consider the more general case of inputs that are finite collections of path-connected compact sets. The main results connect several classical theories such as the conformal change of Riemannian metrics, the theory of positive definite functions of Schoenberg, and screw function theory of Schoenberg and Von Neumann. We develop novel proof techniques based on the combination of screw functions and Lipschitz extensions that may be of independent interest .
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We consider a simple graph-based metric on points in Euclidean space known as the edge-squared metric. This metric is defined by squaring the Euclidean distance between points, and taking the shortest paths on the resulting graph. This metrichas been studied before in wireless networks and machine learning, and has the density sensitive property: distances between two points in the same cluster are short, even if their Euclidean distance is long. This property is desirable
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 the nearest neighbor metric. In fact, we prove the surprising result that a previously published 3-approximation is an exact algorithm. The nearest neighbor metric can be viewed as a special case of a density-based distance used
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. In this paper, we show that this metricis equal to previously studied geodesic-based metric defined on points in Euclidean space. Previous best work showed that these metrics are 3-approximations of each other. It was not known or suspected that these metrics were equal. We give fast algorithms
, or it can be seen as an example of a manifold metric. Previous computational research on such metrics despaired of computing exact distances on account of the apparent difficulty of minimizing over all continuous paths between a pair of points. We leverage the exact computation of the nearest neighbor metric
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of this distance, a problem that can be seen as a generalization of both the Euclidean spanner and
and persistent homology. We also explore the behavior of
R
Euclidean MST problem. Spanners of the edge-squared metric are faster to compute and sparser than the best known Euclidean spanners in a variety of settings
metric built from point sets drawn from an underlying distribution and consider the more general case of inputs that are finite collections of path-connected compact sets. The main results connect several classical theories such as the conformal change of Riemannian metrics, the theory of positive definite functions of Schoenberg, and screw function theory of Schoenberg and Von Neumann. We develop novel proof techniques based on the combination of screw functions and Lipschitz extensions that may be of independent interest