发表机构
National University of Singapore; Nanyang Technological University; Nankai University（新加坡国立大学; 南洋理工大学; 南开大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究提出基于和乐的图曲率离散化框架，引入两种聚合机制实现曲率相关响应的规范等变更新，经单位球校准验证了方法的有效性。
AI 中文摘要
我们提出了一种基于和乐（holonomy）的框架，用于在配备局部对称正定（SPD）度量的图上离散化曲率。每个顶点携带一个纤维度量\textit{g}\textsubscript{\textit{i}}，每条有向边携带一个可逆的、与度量兼容的传输\textit{F}\textsubscript{\textit{ij}}。沿定向三角形环\textit{C}的有序乘积给出和乐\textit{H}\textsubscript{\textit{C}}，其归一化对数\textit{Ω}\textsubscript{\textit{C}}=-s\textsubscript{\textit{C}}\textsuperscript{-1}Log(\textit{H}\textsubscript{\textit{C}})被用作有限环曲率观测值。该构造离散化了“无穷小和乐由曲率控制”的几何原理，而非将和乐视为启发式特征。由于\textit{Ω}\textsubscript{\textit{C}}位于\textit{g}\textsubscript{\textit{i}}正交李代数中，其本身并非SPD度量的速度，因此我们引入两种聚合机制：一是与对称响应矩阵的对易子，产生对称的里奇型度量响应；二是曲率诱导边通量的感知关联的协变散度，反映迹与协变散度的关系。所得响应在局部正交规范下是等变的，可驱动保持正定性的指数更新。我们还给出了边传输的可逆、与度量兼容的参数化，允许在尊重图几何的同时学习正交边因子、环尺度、权重和响应矩阵。在单位球上进行的已知几何校准，验证了和乐-曲率关系、非平凡局部度量表示下的曲率保持，以及从局部观测中经验性恢复边传输的效果。
英文摘要
We propose a holonomy-based framework for discretizing curvature on graphs equipped with local symmetric positive-definite metrics. Each vertex carries a fibre metric \(g_i\), and each directed edge carries a reversible metric-compatible transport \(F_{ij}\). The ordered product around an oriented triangular loop \(\mathcal C\) gives a holonomy \(H_{\mathcal C}\), whose normalized logarithm \(Ω_{\mathcal C}=-s_{\mathcal C}^{-1}\operatorname{Log}(H_{\mathcal C})\) is used as a finite-loop curvature observation. Thus the construction discretizes the geometric principle that infinitesimal holonomy is controlled by curvature, rather than treating holonomy as a heuristic feature. Since \(Ω_{\mathcal C}\) lies in the \(g_i\)-orthogonal Lie algebra, it is not itself a velocity of an SPD metric. We therefore introduce two aggregation mechanisms: a commutator with a symmetric response matrix, producing symmetric Ricci-type metric responses, and an incidence-aware covariant divergence of curvature-induced edge fluxes, reflecting the relation between trace and covariant divergence. The resulting responses are locally orthogonal-gauge equivariant and can drive exponential updates that preserve positive definiteness. We also give a reversible metric-compatible parametrization of edge transports, allowing orthogonal edge factors, loop scales, weights, and response matrices to be learned while respecting the graph geometry. Known-geometry calibrations on the unit sphere test the holonomy--curvature relation, curvature preservation under nontrivial local metric representations, and the empirical recovery of edge transports from local observations.
