I-BBS: Coordinate-Free Inference of Latent Sub-Manifolds Using Random Distance Matrix Theory
2026-06-29 • Machine Learning
Machine Learning
AI summaryⓘ
The authors, Bogomolny, Bohigas, and Schmit, showed that the pattern of distances between points sampled from a smooth shape (manifold) reveals information about that shape. Building on this, the authors here developed I-BBS, a method that finds the low-dimensional shape hidden inside high-dimensional distance data without needing actual coordinate information. They handle noise that mixes true shape data with irrelevant parts by using stable integer patterns from the spectrum rather than individual eigenvalues. Tests on synthetic spheres show this method can reliably find both the shape's dimension and noise characteristics from just one distance matrix.
ManifoldPairwise distance matrixEigenvaluesLatent sub-manifoldHigh-dimensional dataNoise modelsSpectral analysisPerron multipletDimensionality inferenceGenerative noise
Authors
Igor Halperin
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.