用于神经元表征的热带代数几何：一种基于Arakelov-Green测度的图学习描述符
CLEARMIND: Closed-Form Laplacian Embeddings of Arakelov-Green Resistance for Morphological Intrinsic Neuronal Distances
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中文总结 AI 辅助
该研究针对三维神经元形态分析的图学习问题，提出基于热带代数几何的无训练几何先验，构建Arakelov-Green测度相关描述符，在多数据集上提升了分类性能。
中文摘要 AI 辅助
对三维神经元形态的定量分析需要同时捕捉图拓扑结构和空间几何特征。当前的消息传递图神经网络（GNN）受限于1-Weisfeiler-Lehman（1-WL）测试，限制了其捕捉空间邻近性诱导的环的能力。为解决该问题，我们提出一种基于热带代数几何的无训练几何先验。我们将新近建立的热带Abel-Jacobi变换和极化距离应用于树状数据的机器学习，引入包含环空间增强和商空间构建的结构变换流水线，以将空间树转换为适合嵌入热带雅可比的环度量图。计算精确的热带极化距离需要求解整数格上的NP难最近向量问题（CVP），我们未采用带量化误差的显式近似（如Babai舍入），而是在Albanese环的通用覆盖上采用连续松弛。我们证明，通过图拉普拉斯算子的广义逆以闭式计算的离散Arakelov-Green测度可精确分解为内在路径度量减去该覆盖上未量化的极化距离，从而避免整数格搜索。该度量产生两种描述符：特征向量提供节点级结构坐标，置换不变的特征值谱提供图级签名。在BREC基准上，特征向量公式展现出超越1-WL极限的表达能力；在三维形态数据集（ACT-4、JML-4、BIL-6）上，该谱可无缝集成到标准架构（VAEs、GNNs、Tree-LSTMs）中，无需额外可训练参数，性能优于显式格近似，且相比现有空间模型提升了分类准确率。
英文摘要
Tropical geometry turns a metric graph into a flat torus, its tropical Jacobian, and measures distances between the images of its points by a closest vector problem on the period lattice. We prove that for points of the graph this problem is solved by geodesics: the squared tropical polarization distance equals the path metric minus the Arakelov-Green (AG) distance, an effective-resistance kernel given in closed form by the Laplacian pseudoinverse. Equivalently, the AG distance and the path metric are the minimal energies of real and of integral unit flows, and the squared tropical distance is exactly their integrality gap. The full tropical distance matrix is thus computable in cubic time. On this identity we build CLEARMIND, a training-free descriptor of 3D neuronal morphology. A reconstruction is reduced to its branching skeleton, neurite tips near the soma are joined to it, short bridges are contracted, and the AG matrix of the result is summarized by its spectrum. Every step has an exact algebraic description; for instance, the period matrix records the shared soma-to-tip path lengths, and the spectrum has exactly one positive eigenvalue, so its absolute values determine it. The signature needs no lattice search, is invariant to vertex order, rigid motions and subdivision, and is Lipschitz in the edge lengths. On ACT-4, JML-4 and BIL-6, appending the 64-dimensional spectrum improves a point-cloud GNN and MorphVAE on every dataset, by up to 20.3 points; a Tree-LSTM with the spectrum outperforms every reimplemented baseline; and an MLP on the spectrum alone is a competitive classifier that surpasses every reimplemented deep baseline on ACT-4. On BREC, a GIN with AG eigenvector encodings distinguishes 70.0% of the pairs beyond the 1-WL limit, including 33% of the CFI pairs, more than PPGN (23%) and I$^2$-GNN (21%), and the spectrum alone separates 217 of 400 pairs without training.
