Some researchers have proposed using non-Euclidean metrics for clustering data points . Generally, the metric should recognize that two points in the same cluster are close , even if their Euclidean distance is far. Multiple proposals have been suggested, including the Edge-Squared Metric (a specific example of a graph geodesic) and the Nearest Neighbor Metric, a geodesic metric . In this paper, we prove that the edge-squared and nearest-neighbor metrics are in fact equivalent . Previous best work showed that  the  edge-squared metric was a 3-approximation of the Nearest Neighbor metric. This paper represents one of the first proofs of equating a continuous metric with a discrete metric, using non-trivial discrete methods. Our proof uses the Kirszbraun theorem (also known as the Lipschitz Extension Theorem and Brehm's Extension Theorem), a notable theorem in functional analysis and computational geometry. The results of our paper, combined with the results of Hwang, Damelin, and Hero, tell us that the Nearest Neighbor distance on i. i.d samples of a density is a reasonable constant approximation of a natural density-based distance function .
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 metric has 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 metric is 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 .
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Some researchers have proposed using non-Euclidean metrics for clustering data points . Generally, the metric should recognize that
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 metric has been studied before in wireless networks and machine learning, and has the density sensitive property: distances between
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close
short
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far. Multiple proposals have been suggested, including the Edge-Squared Metric (a specific example of a graph geodesic) and the Nearest Neighbor Metric, a geodesic metric
long. This property is desirable in machine learning
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prove that the edge-squared and nearest-neighbor metrics are in fact equivalent
show that this metric is equal to previously studied geodesic-based metric defined on points in Euclidean space
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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
A
Euclidean MST problem. Spanners of the
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metric was a 3-approximation of the Nearest Neighbor metric. This paper represents one of the first proofs of equating a continuous metric with a discrete metric, using non-trivial discrete methods. Our proof uses the Kirszbraun theorem (also known as the Lipschitz Extension Theorem and Brehm's Extension Theorem), a notable theorem in functional analysis and computational geometry. The results of our paper, combined with the results of Hwang, Damelin, and Hero, tell us that the Nearest Neighbor distance on i. i.d samples of a density is a reasonable constant approximation of a natural density-based distance function
metric are faster to compute and sparser than the best known Euclidean spanners in a variety of settings