Composition of nested embeddings with an application to outlier removal

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 $O(\log^2 k)$ factor and the distortion to within a constant factor. The main technical component in our result is an approach for constructing Lipschitz extensions of embeddings into Banach spaces (such as $\ell_p$ spaces). We consider a stronger version of Lipschitz extension that we call a \textit{nested composition of embeddings}: given a low distortion embedding of a subset $S$ of the metric space $X$, our goal is to extend this embedding to all of $X$ such that the distortion over $S$ is preserved, whereas the distortion over the remaining pairs of points in $X$ is bounded by a function of the size of $X\setminus S$. Prior work on Lipschitz extension considers settings where the size of $X$ is potentially much larger than that of $S$ and the expansion bounds depend on $|S|$. In our setting, the set $S$ is nearly all of $X$ and the remaining set $X\setminus S$, a.k.a. the outliers, is small. We achieve an expansion bound that is logarithmic in $|X\setminus S|$.