I-BBS: Coordinate-Free Inference of Latent Sub-Manifolds Using Random Distance Matrix Theory

2026-06-28

AI Summary

This study addresses the challenge of identifying a low-dimensional latent submanifold solely from a high-dimensional distance matrix, without access to the original coordinate data. Building on Bogomolny–Bohigas–Schmit spectral theory, the authors propose two integer-valued, noise-robust spectral signatures: one leverages the multiplicities of non-Perron eigenvalues to precisely estimate the intrinsic manifold dimension and reveals its scaling law with respect to noise intensity, thereby overcoming the instability inherent in conventional continuous-spectrum slope methods. By integrating random matrix theory, spectral analysis, and generative noise modeling—encompassing both model-driven and model-free approaches—the method accurately recovers both the true manifold dimension and the underlying noise type from synthetic noisy data on S¹, S², and S³, achieving joint blind identification of manifold structure and noise characteristics from a single distance matrix.

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 $S^1$, $S^2$ and $S^3$ 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.

Problem

• latent sub-manifold
• distance matrix
• coordinate-free inference
• random matrix theory
• manifold learning

Innovation

• coordinate-free inference
• latent sub-manifold
• random distance matrix
• spectral signatures
• noise-robust geometry