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
该研究针对分支点处的加权切几何恢复问题，基于单尺度得分场建立切测度模型，通过线性方程组校准等方法实现几何参数与测度的恢复，在平面情形中验证了方法的有效性。
AI 中文摘要
在光滑数据流形附近，一个切空间即可概括局部几何；而在分支点处，对应的一阶对象变为切方向上的测度，其归一化质量记录了所选数据测度下各分支的局部占比。我们提出问题：当分支中心和齐次度$d$未知时，某一噪声水平下的得分场是否能确定该加权切几何。在这一切测度模型中，$d$为局部测度维数。齐次切测度的高斯平滑满足Ornstein--Uhlenbeck本征函数方程，其弱形式将得分值（无需得分导数）转化为关于中心和齐次度的线性方程组，具有明确的秩条件和扰动界。经此校准后，某一球面上的切向得分是标量高斯-锥变换的球对数梯度，积分可恢复该变换（比例因子除外），且其所有球谐乘数均为正，因此任意环境维数$D\geq2$下，一个精确的球壳即可确定归一化角测度。对于至多$K$条正射线，通过次数$2K-1$的矩可在任意维数下构造性地恢复计数、方向和权重；任意固定观测方案至少需要$KD-1$个标量切向分量。在平面情形中，次数$K$是充分且必要的，我们给出了定量的有限查询证书。对于具有正$C^{0,\beta}$密度的有限平面$C^{1,\beta}$分支，我们证明有限噪声得分到其切模型的收敛速度为$O(\sigma^\beta)$。在控制实验中，5万步训练降低了四种几何结构的验证归一化得分误差，但增大了角动量误差，区分了普通得分拟合与几何恢复。
英文摘要
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 $D\geq2$. 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 $C^{1,β}$ branches with positive $C^{0,β}$ densities, we prove $O(σ^β)$ 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.
