Local Distance Graph Yields Global Euclidean Embedding

A new method recovers a globally consistent Euclidean embedding of data using only local pairwise distances. The approach operates on a neighborhood graph without prior vector representations, solving a variational problem that matches local graph distances to the Euclidean metric via embedding function differentials. Euler-Lagrange equations are derived in coordinate-free form, enabling direct operator evaluation from the distance graph. Although nonlinear, the equations reduce to an iteratively updated sparse linear system. The main contribution is the derivation of functional equations governing the optima.