I-BBS: Coordinate-Free Inference of Latent Sub-Manifolds Using Random Distance Matrix Theory
Igor Halperin
cs.LG, cond-mat.dis-nn, stat.ML
Submitted: 2026-06-29
Comments: 53 pages, 23 figures
Code: https://github.com/ighalp/I-BBS
License: http://creativecommons.org/licenses/by-nc-nd/4.0/
The gist: Bogomolny, Bohigas and Schmit (BBS) found that the spectrum of the pairwise distance matrix on N points sampled from a smooth d-dimensional manifold encodes a signature of the underlying geometry.
Terminology
Abstract
Bogomolny, Bohigas and Schmit (BBS) found that the spectrum of the pairwise distance matrix on N points sampled from a smooth d-dimensional manifold encodes a signature of the underlying geometry. We develop I-BBS (Inference-BBS), a coordinate-free method that identifies a low-dimensional latent sub-manifold embedded in a high-dimensional ambient distance matrix alone, without accessing an ambient high-dimensional vector space. It therefore applies even when that space is only partly observable or undefined. We model the ambient embedding by two classes of generative noise, model-based and model-free. The noise mixes the latent signal with off-manifold components, so the eigenvalues reorganise collectively and the latent geometry cannot be read off eigenvalue by eigenvalue. We recover it instead from two integer-stable signatures that survive the noise: the multiplicity of the top non-Perron multiplet, which fixes d, and a parameter-free law for how the multiplet positions shrink as the noise grows. On synthetic spheres S 1, S squared and S cubed these integer signatures are far more stable under noise than the continuous spectral slope, and a blind test recovers both the manifold and the noise model from a single distance matrix. Applications to neural-network representations and to the dynamic training regime are developed in two companion papers.
Sources
- Spectral properties of distance matrices
- Distance matrices and isometric embeddings
- On Euclidean random matrices in high dimension
- Largest eigenvalue and top eigenvector statistics of large Euclidean random matrices
- Statistical Inference for Manifold Similarity and Alignability across Noisy High-Dimensional Datasets
- Random Features for Kernel Approximation: A Survey on Algorithms, Theory, and Beyond
- Euclidean random matrices: solved and open problems
- Euclidean random matrices and their applications in physics
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