# Lossless Prioritized Embeddings

## TL;DR

This paper introduces two lossless prioritized embeddings for metric spaces, achieving exact preservation of distances for specific points and improving prior methods by eliminating distortion loss.

## Contribution

It presents the first lossless prioritized embeddings for tree metrics and general metrics into - , matching classical worst-case guarantees.

## Key findings

- Isometric prioritized embedding of tree metrics into  with dimension O(log j)
- Prioritized Matousek's embedding with distortion depending on j and n
- Optimal distortion embeddings into ultra-metric and spanning trees

## Abstract

Given metric spaces $(X,d)$ and $(Y,\rho)$ and an ordering $x_1,x_2,\ldots,x_n$ of $(X,d)$, an embedding $f: X \rightarrow Y$ is said to have a prioritized distortion $\alpha(\cdot)$, if for any pair $x_j,x'$ of distinct points in $X$, the distortion provided by $f$ for this pair is at most $\alpha(j)$. If $Y$ is a normed space, the embedding is said to have prioritized dimension $\beta(\cdot)$, if $f(x_j)$ may have nonzero entries only in its first $\beta(j)$ coordinates. The notion of prioritized embedding was introduced by \cite{EFN15}, where a general methodology for constructing such embeddings was developed. Though this methodology enables \cite{EFN15} to come up with many prioritized embeddings, it typically incurs some loss in the distortion. This loss is problematic for isometric embeddings. It is also troublesome for Matousek's embedding of general metrics into $\ell_\infty$, which for a parameter $k = 1,2,\ldots$, provides distortion $2k-1$ and dimension $O(k \log n \cdot n^{1/k})$. In this paper we devise two lossless prioritized embeddings. The first one is an isometric prioritized embedding of tree metrics into $\ell_\infty$ with dimension $O(\log j)$. The second one is a prioritized Matousek's embedding of general metrics into $\ell_\infty$, which provides prioritized distortion $2 \lceil k {{\log j} \over {\log n}} \rceil - 1$ and dimension $O(k \log n \cdot n^{1/k})$, again matching the worst-case guarantee $2k-1$ in the distortion of the classical Matousek's embedding. We also provide a dimension-prioritized variant of Matousek's embedding. Finally, we devise prioritized embeddings of general metrics into (single) ultra-metric and of general graphs into (single) spanning tree with asymptotically optimal distortion.