Metric embedding with outliers

We initiate the study of metric embeddings with \emph{outliers}. Given some metric space $(X,ρ)$ we wish to find a small set of outlier points $K \subset X$ and either an isometric or a low-distortion embedding of $(X\setminus K,ρ)$ 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 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 $O(n^2)$ 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 $d\geq 2$ 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 $\ell_{\infty}$ distortion. We present polynomial-time \emph{bi-criteria} approximation algorithms. Specifically, given some $ε> 0$, let $k_ε$ 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 $3k_ε$ outliers and an embedding of the remaining points into an ultrametric with distortion $O(ε\log n)$. 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 $2k_ε$ outliers and an embedding of the remaining points with distortion $O(\sqrtε)$.

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