# Euclidean Embedding of Data Using Local Distances

arXiv: https://arxiv.org/abs/2605.19243 · HTML: https://mlsanity.com/paper/2605.19243/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 4 of its 20 references that are in the map.
- First posted on arXiv 19 May 2026.

## Builds on (most cited first, 4 of 4)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Estimating the intrinsic dimension of datasets by a minimal neighborhood information](https://mlsanity.com/paper/1803.06992/) (2018)
- [Visualizing Large-scale and High-dimensional Data](https://mlsanity.com/paper/1602.00370/) (2016)
- [Minimum-Distortion Embedding](https://mlsanity.com/paper/2103.02559/) (2021)

## Abstract (arXiv)

We study the problem of recovering a globally consistent Euclidean embedding of data, given only a local distance graph and propose a method that optimally represents these distances. The method operates solely on a neighborhood graph weighted by pairwise distances, without requiring any prior vector representation of the data. The embedding is obtained by solving a variational problem that matches local, on-graph distances to the Euclidean metric, induced by the differentials of the embedding functions. The resulting Euler-Lagrange equations are derived in a coordinate-free form, enabling direct evaluation of all operators from the distance graph alone. Though non-linear and missing an explicit expression for their non-linearity, these equations are shown to be resolved as an iteratively updated sparse linear problem. The main contributions of the proposed approach are (a) the derivation of the functional equations governing the optimal Euclidean embedding in the continuum, (b) a representation-free formulation that requires only a neighborhood distance graph and no feature vectors and (c) an estimation procedure based exclusively on local graph operations. We experimentally evaluate the resulting non-parametric algorithm on synthetic manifolds and real datasets, demonstrating consistent preservation of local metric structure and neighboring relations, while approximating the global isometric embedding.