# Matched Queries for Curvature and Density at Branching Junctions

arXiv: https://arxiv.org/abs/2609.01319 · HTML: https://mlsanity.com/paper/2609.01319/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 25 references that are in the map.
- First posted on arXiv 1 Sep 2026.

## Builds on (most cited first, 9 of 9)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Manifolds, Random Matrices and Spectral Gaps: The geometric phases of generative diffusion](https://mlsanity.com/paper/2410.05898/) (2024)
- [When Scores Learn Geometry: Rate Separations under the Manifold Hypothesis](https://mlsanity.com/paper/2509.24912/) (2025)
- [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)
- [Boundary-layer asymptotics for Gaussian-smoothed singular measures](https://mlsanity.com/paper/2607.04514/) (2026)
- [Rao-Blackwellized Score Matching on Manifolds](https://mlsanity.com/paper/2605.25567/) (2026)
- [Recovering Weighted Tangent Geometry from a Single-Scale Score Field](https://mlsanity.com/paper/2608.22334/) (2026)

## Abstract (arXiv)

At a junction, a score field can reveal weighted tangent rays, yet these first-order quantities do not determine how individual branches bend or how their densities change away from the center. Recovering this missing information is necessary for describing local continuation beyond a single point, but finite observations must separate branchwise second-order effects while allowing error in the estimated center. We address this inverse problem using matched score queries at noise scales σ and λσ. For a finite union of C2,α half-branches in ℝD, the normalized score has the expansion Fσ=F0+σ G+O(σ1+α). Matched subtraction cancels the tangent contribution and exposes G, which depends linearly on branchwise curvature and log-density slope. Given tangent directions and weights on distinct rays, G uniquely identifies all sD branch parameters, and sD scalar component observations are necessary. An O(σ2) center error introduces D translation modes, leading to (s+1)D observations under full-rank calibration, except for a translation-invariant full line. We also establish a perturbation bound and a conditional kernel-density-estimation rate. Experiments reproduce the predicted population and N-1/5 trends and remain full rank up to D=20 with 16 supplied branches. In end-to-end tests for D=3--5, a known-count first-order frontend yields full rank in all 135 population systems and a median relative jet error of 0.132. With strong first-order error, matched responses reduce median parameter error by a factor of 49.4 relative to naive tangent subtraction.