发表机构
School of Computer Science and Engineering; Southeast University（计算机科学与工程学院; 东南大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究针对分支节点处曲率与密度的逆问题，提出利用匹配得分查询的方法，通过实验验证了该方法在高维场景下的有效性，可显著降低参数误差。
AI 中文摘要
在节点处，得分场可揭示加权切向射线，但这些一阶量无法确定各分支如何弯曲，也无法确定它们的密度如何远离中心变化。恢复这些缺失信息对于描述单点之外的局部延续性是必要的，但有限观测必须分离分支方向的二阶效应，同时允许估计中心存在误差。我们利用噪声尺度为σ和λσ的匹配得分查询来解决这一逆问题。对于ℝ^D中有限个C^{2,α}半分支的并集，归一化得分具有展开式F_σ=F_0+σG+O(σ^{1+α})。匹配相减可抵消切向贡献并揭示G，G线性依赖于分支方向的曲率和对数密度斜率。给定不同射线上的切向方向和权重，G可唯一识别所有sD个分支参数，且需要sD个标量分量观测值。O(σ^2)的中心误差会引入D个平移模式，在满秩校准下会产生(s+1)D个观测值，平移不变的完整直线除外。我们还建立了一个扰动界和条件核密度估计速率。实验重现了预测的总体趋势和N^{-1/5}趋势，在提供16个分支的情况下，直至D=20时仍保持满秩。在D=3至5的端到端测试中，已知计数的一阶前端在所有135个总体系统中均产生满秩，且中位相对射流误差为0.132。在存在强一阶误差的情况下，与朴素切向相减相比，匹配响应将中位参数误差降低了49.4倍。
英文摘要
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 $C^{2,α}$ half-branches in $\mathbb{R}^D$, the normalized score has the expansion $F_σ=F_0+σG+O(σ^{1+α})$. 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 $O(σ^2)$ 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.
