Matched Queries for Curvature and Density at Branching Junctions

Abstract: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 $\sigma$ and $\lambda\sigma$. For a finite union of $C^{2,\alpha}$ half-branches in $\mathbb{R}^D$, the normalized score has the expansion $F_\sigma=F_0+\sigma G+O(\sigma^{1+\alpha})$. 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 $O(\sigma^2)$ 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.

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

arXiv:2609.01319 [stat.ML]

(or arXiv:2609.01319v1 [stat.ML] for this version)

https://doi.org/10.48550/arXiv.2609.01319 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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