🤖 AI Summary
This study addresses the problem of recovering a globally consistent Euclidean embedding from local distance graphs alone. To this end, the authors propose a purely graph-based distance modeling framework that operates without access to original features, relying solely on neighborhood distance information. By employing a variational approach, the method aligns local graph distances with the Euclidean metric induced by the embedding. Key contributions include the formulation of a functional equation characterizing Euclidean embeddings in the continuous domain, the derivation of a coordinate-invariant Euler–Lagrange equation, and the design of a sparse linear iterative scheme for localized computation. Experiments demonstrate that the proposed approach effectively preserves local metric structure and neighborhood relationships on both synthetic manifolds and real-world datasets, achieving embeddings that closely approximate global isometries.
📝 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.