Metric embedding with outliers

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 $(X,\rho)$ we wish to find a small set of outlier points $K \subset X$ and either an isometric or a low-distortion embedding of $(X\setminus K,\rho)$ 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 $\epsilon > 0$, let $k_\epsilon$ denote the minimum number of outliers required to obtain an embedding with distortion $\epsilon$. For the case of embedding into ultrametrics we obtain a polynomial-time algorithm which computes a set of at most $3k_{\epsilon}$ outliers and an embedding of the remaining points into an ultrametric with distortion $O(\epsilon \log n)$. 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 $2k_{\epsilon}$ outliers and an embedding of the remaining points with distortion $O(\sqrt{\epsilon})$.