Metric embedding with outliers
Anastasios Sidiropoulos, Yusu Wang

TL;DR
This paper studies metric embeddings with outliers, proposing algorithms for minimizing outliers in isometric embeddings into ultrametrics, trees, and Euclidean spaces, and providing NP-hardness results and bi-criteria approximation algorithms for small distortion embeddings.
Contribution
It introduces polynomial-time algorithms for outlier minimization in isometric embeddings and develops bi-criteria approximation algorithms for low-distortion embeddings with outliers.
Findings
Polynomial-time approximation algorithms with factors 3, 4, and 2 for ultrametrics, trees, and Euclidean spaces.
Optimal $O(n^2)$ algorithms for ultrametric and tree embeddings.
NP-hardness and inapproximability results for outlier embeddings into ultrametrics, trees, and Euclidean spaces.
Abstract
We initiate the study of metric embeddings with \emph{outliers}. Given some metric space we wish to find a small set of outlier points and either an isometric or a low-distortion embedding of into some target metric space. This is a natural problem that captures scenarios where a small fraction of points in the input corresponds to noise. For the case of isometric embeddings we derive polynomial-time approximation algorithms for minimizing the number of outliers when the target space is an ultrametric, a tree metric, or constant-dimensional Euclidean space. The approximation factors are 3, 4 and 2, respectively. For the case of embedding into an ultrametric or tree metric, we further improve the running time to for an -point input metric space, which is optimal. We complement these upper bounds by showing that outlier…
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Taxonomy
TopicsComputational Geometry and Mesh Generation · Complexity and Algorithms in Graphs · Advanced Numerical Analysis Techniques
