We study the design of embeddings into Euclidean space with outliers. Given a metric space $(X,d)$ and an integer $k$, the goal is to embed all but $k$ points in $X$ (called the "outliers") into $\ell_2$ with the smallest possible distortion $c$. Finding the optimal distortion $c$ for a given outlier set size $k$, or alternately the smallest $k$ for a given target distortion $c$ are both NP-hard problems. In fact, it is UGC-hard to approximate $k$ to within a factor smaller than $2$ even when the metric sans outliers is isometrically embeddable into $\ell_2$. We consider bi-criteria approximations. Our main result is a polynomial time algorithm that approximates the outlier set size to within an $O(\log^4 k)$ factor and the distortion to within a constant factor. The main technical component in our result is an approach for constructing a composition of two given embeddings from subsets of $X$ into $\ell_2$ which inherits the distortions of each to within small multiplicative factors. Specifically, given a low $c_S$ distortion embedding from $S\subset X$ into $\ell_2$ and a high(er) $c_X$ distortion embedding from the entire set $X$ into $\ell_2$, we construct a single embedding that achieves the same distortion $c_S$ over pairs of points in $S$ and an expansion of at most $O(\log k)\cdot c_X$ over the remaining pairs of points, where $k=|X\setminus S|$. Our composition theorem extends to embeddings into arbitrary $\ell_p$ metrics for $p\ge 1$, and may be of independent interest. While unions of embeddings over disjoint sets have been studied previously, to our knowledge, this is the first work to consider compositions of nested embeddings.

翻译：我们研究具有离群点的欧几里得空间嵌入的设计问题。给定度量空间 $(X,d)$ 和整数 $k$，目标是除 $X$ 中 $k$ 个点（称为"离群点"）外，将所有点嵌入 $\ell_2$ 空间，并使得失真 $c$ 尽可能小。对于给定的离群点集大小 $k$ 寻找最优失真 $c$，或对于给定目标失真 $c$ 寻找最小 $k$，均为 NP 难问题。事实上，即使无离群点的度量空间可等距嵌入 $\ell_2$，在唯一博弈难（UGC-hard）假设下，$k$ 的近似因子也无法小于 $2$。我们考虑双准则近似。主要成果是一个多项式时间算法，该算法在 $O(\log^4 k)$ 因子内近似离群点集大小，并在常数因子内逼近失真。本成果的核心技术组件是一种方法，用于构建 $X$ 子集到 $\ell_2$ 的两个给定嵌入的组合，该组合在较小的乘法因子内继承每个嵌入的失真性质。具体而言，给定从 $S\subset X$ 到 $\ell_2$ 的低失真 $c_S$ 嵌入，以及从整个集合 $X$ 到 $\ell_2$ 的较高失真 $c_X$ 嵌入，我们构建一个单一嵌入，使得在 $S$ 中任意点对上保持相同失真 $c_S$，而在其余点对上实现至多 $O(\log k)\cdot c_X$ 的扩张，其中 $k=|X\setminus S|$。我们的组合定理可推广到 $p\ge 1$ 的任意 $\ell_p$ 度量嵌入，并可能具有独立研究价值。尽管此前已有关于不相交集合上嵌入并集的研究，但据我们所知，这是首个针对嵌套嵌入组合的工作。