# Recovering Weighted Tangent Geometry from a Single-Scale Score Field

arXiv: https://arxiv.org/abs/2608.22334 · HTML: https://mlsanity.com/paper/2608.22334/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 30 references that are in the map.
- First posted on arXiv 23 Aug 2026.

## Builds on (most cited first, 15 of 15)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Kernel Distribution Embeddings: Universal Kernels, Characteristic Kernels and Kernel Metrics on Distributions](https://mlsanity.com/paper/1604.05251/) (2016)
- [Manifolds, Random Matrices and Spectral Gaps: The geometric phases of generative diffusion](https://mlsanity.com/paper/2410.05898/) (2024)
- [Improving the Euclidean Diffusion Generation of Manifold Data by Mitigating Score Function Singularity](https://mlsanity.com/paper/2505.09922/) (2025)
- [When Scores Learn Geometry: Rate Separations under the Manifold Hypothesis](https://mlsanity.com/paper/2509.24912/) (2025)
- [Topological Singularity Detection at Multiple Scales](https://mlsanity.com/paper/2210.00069/) (2022)
- [Landing with the Score: Riemannian Optimization through Denoising](https://mlsanity.com/paper/2509.23357/) (2025)
- [Diffusion Model for Manifold Data: Score Decomposition, Curvature, and Statistical Complexity](https://mlsanity.com/paper/2603.20645/) (2026)
- [Local Coverage Governs Memorization in Diffusion Models](https://mlsanity.com/paper/2606.14390/) (2026)
- [HADES: Fast Singularity Detection with Local Measure Comparison](https://mlsanity.com/paper/2311.04171/) (2023)
- [Boundary-layer asymptotics for Gaussian-smoothed singular measures](https://mlsanity.com/paper/2607.04514/) (2026)
- [Diffusion models recover accurate mixture weights despite score function insensitivity](https://mlsanity.com/paper/2607.15485/) (2026)
- [Multi-Subspace Multi-Modal Modeling for Diffusion Models: Estimation, Convergence and Mixture of Experts](https://mlsanity.com/paper/2601.01475/) (2026)
- [Guiding Vector Field Generation via Score-based Diffusion Model](https://mlsanity.com/paper/2604.24487/) (2026)
- [Score Approximation for Diffusion Models on Arbitrary Low-Dimensional Structures](https://mlsanity.com/paper/2606.19894/) (2026)

## Built on by (most cited first, 1 of 1)

- [Matched Queries for Curvature and Density at Branching Junctions](https://mlsanity.com/paper/2609.01319/) (2026)

## Abstract (arXiv)

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≥2. 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 C1,β branches with positive C0,β densities, we prove O(σβ) 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.