Recovering Weighted Tangent Geometry from a Single-Scale Score Field
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 ...
| Authors | Ziqi Zhao, Qingjian Ni |
|---|---|
| Primary category | stat.ML |
| Published | 23 August 2026 |
| Last updated at source | 23 August 2026 |
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