I-BBS: Coordinate-Free Inference of Latent Sub-Manifolds Using Random Distance Matrix Theory
Provides a coordinate-free method for manifold inference from distance data, applicable when ambient space is unavailable or noisy, with potential impact on analyzing neural representations.
I-BBS infers the dimension and geometry of a latent sub-manifold from a distance matrix alone, without requiring coordinates. It recovers integer-stable signatures (multiplicity of top eigenvalues and a noise-scaling law) that are robust to noise, demonstrated on synthetic spheres S1, S2, S3.
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.