Metric embedding with outliers

We initiate the study of metric embeddings with outliers. Given some metric space (X, ρ) we wish to find a small set of outlier points K ⊂ X and either an isometric or a low-distortion embedding of (X \K, ρ) 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 some 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 O(n) for an n-point input metric space, which is optimal. We complement these upper bounds by showing that outlier embedding into ultrametrics, trees, and d-dimensional Euclidean space for any d ≥ 2 are all NP-hard, as well as NP-hard to approximate within a factor better than 2 assuming the Unique Game Conjecture. For the case of non-isometries we consider embeddings with small `∞ distortion. We present polynomial-time bi-criteria approximation algorithms. Specifically, given some ε > 0, let kε denote the minimum number of outliers required to obtain an embedding with distortion ε. For the case of embedding into ultrametrics we obtain a polynomial-time algorithm which computes a set of at most 3kε outliers and an embedding of the remaining points into an ultrametric with distortion O(ε log n). Finally, for embedding a metric of unit diameter into constant-dimensional Euclidean space we present a polynomial-time algorithm which computes a set of at most 2kε outliers and an embedding of the remaining points with distortion O( √ ε). ∗Dept. of Computer Science and Engineering and Dept. of Mathematics, The Ohio State University. Columbus, OH, USA. Supported by NSF grants CCF 1423230 and CAREER 1453472. †Dept. of Computer Science and Engineering, The Ohio State University. Columbus, OH, USA. The work is partially supported by NSF under grant CCF-1319406. ar X iv :1 50 8. 03 60 0v 1 [ cs .D S] 1 4 A ug 2 01 5