Recovering Weighted Tangent Geometry from a Single-Scale Score Field

Abstract:Near a smooth data manifold, one tangent space summarizes local geometry. At a branch point, the corresponding first-order object is instead a measure over tangent directions, whose normalized masses record the local share of each branch under the chosen data measure. We ask whether a score field at one noise level determines this weighted tangent geometry when the branch center and homogeneity degree $d$ are unknown. In this tangent-measure model, $d$ is the local measure dimension. Gaussian smoothing of a homogeneous tangent measure satisfies an Ornstein--Uhlenbeck eigenfunction equation. Its weak form turns score values---without score derivatives---into a linear system for the center and homogeneity degree, with an explicit rank condition and perturbation bound. After this calibration, the tangential score on one sphere is the spherical log-gradient of a scalar Gaussian--cone transform. Integration recovers that transform up to scale, and all its spherical-harmonic multipliers are positive. Thus one exact shell identifies the normalized angular measure in every ambient dimension $D\geq2$. For at most $K$ positive rays, moments through degree $2K-1$ constructively recover count, directions, and weights in arbitrary dimension. Any fixed observation scheme needs at least $KD-1$ scalar tangential components. In the plane, degree $K$ is both sufficient and necessary, and we give quantitative finite-query certificates. For finite planar $C^{1,\beta}$ branches with positive $C^{0,\beta}$ densities, we prove $O(\sigma^\beta)$ convergence from the finite-noise score to its tangent model. In controlled experiments, 50k-step training lowers validation normalized-score error across four geometries yet raises angular-moment error, separating ordinary score fit from geometry recovery.

Machine Learning (stat.ML); Machine Learning (cs.LG)

arXiv:2608.22334 [stat.ML]

(or arXiv:2608.22334v1 [stat.ML] for this version)

https://doi.org/10.48550/arXiv.2608.22334 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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