Euclidean Embedding of Data Using Local Distances

TL;DR

This paper introduces a novel method for recovering a Euclidean embedding from local distance graphs, using a variational approach that relies solely on local distances without prior data vectors.

Contribution

It derives functional equations for optimal embeddings, offers a feature-free formulation based on local graph operations, and presents an iterative linear solution method.

Findings

The method accurately preserves local metric structure.

It approximates global isometric embeddings on synthetic and real data.

The approach requires only local distance information and no feature vectors.

Abstract

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.