从信息几何到喷注子结构：累积张量、能量相关子和超图的三元关系
From Information Geometry to Jet Substructure: A Triality of Cumulant Tensors, Energy Correlators, and Hypergraphs
AI总结：
本文提出通过高阶Fisher张量解决传统 pairwise Fisher 图无法区分多观测辐射模式的问题，通过局部指数族嵌入构建物理信息超图，并展示其在喷注子结构、观测基设计等应用中的有效性。
摘要：
Pairwise Fisher graphs 能捕捉局部协方差信息，但无法区分不可约多观测辐射模式与普通成对相关子的集合。本文显示，这种缺失的结构可通过高阶Fisher张量自然补充。在有限的binning EECs、ECFs或EFPs基底中，以及由该基底生成的自然指数族坐标中，相同的局部张量有三种等价解释：局部Kullback-Leibler展开的系数、所选相关子观测量的连接累积量，以及连接这些观测量的超边的符号权重。这提供了局部指数族嵌入中的精确Fisher相关子超图三元关系。三元关系为从相关子数据直接构建物理信息超图提供了直接构造。将二次Fisher矩阵扩展到第一个非平凡高阶张量，能够识别真正的连接多观测辐射模式，提供超边权重用于高阶拉普拉斯和消息传递，并给出压缩观测基的原理性标准。本文开发了这些构造并阐述了精确累积量解释为何特殊于自然指数族坐标。在四个应用中展示了该框架。在最小的局部-KL研究中，立方Fisher张量减少了KL截断误差并隔离了主导的三元结构。在两对三喷注子结构基准中，超图选择器改进了压缩基分类。在33个观测基设计问题中，Fisher超图在十二个观测中保留了更多的三阶局部响应。一个低容量学习基准则展示了如何将相同的Fisher超边用作可解释的归纳偏置用于相关子观测量的消息传递。
英文摘要：
Pairwise Fisher graphs capture local covariance information, but they cannot distinguish an irreducible multi-observable radiation pattern from a collection of ordinary pairwise correlations. We show that this missing structure is naturally supplied by higher-order Fisher tensors. In a finite basis of binned EECs, ECFs, or EFPs, and in the natural exponential-family coordinates generated by that basis, the same local tensor has three equivalent interpretations: a coefficient in the local Kullback-Leibler expansion, a connected cumulant of the chosen correlator observables, and a signed weight on a hyperedge linking those observables. This gives an exact Fisher-correlator-hypergraph triality in the local exponential-family embedding. The triality provides a direct construction of physics-informed hypergraphs from correlator data. Extending the quadratic Fisher matrix to the first non-trivial higher tensor identifies genuinely connected multi-observable radiation patterns, supplies hyperedge weights for higher-order Laplacians and message passing, and gives a principled criterion for compressing observable bases beyond pairwise information. We develop these constructions and spell out why the exact cumulant interpretation is special to natural exponential-family coordinates. We illustrate the framework in four applications. In a minimal local-KL study, the cubic Fisher tensor reduces the KL truncation error and isolates the dominant triplet structure. In a two-versus-three prong jet substructure benchmark, the hypergraph selector improves compressed-basis classification. In a 33-observable basis-design problem, the Fisher hypergraph retains more third-order local response at twelve observables. A low-capacity learning benchmark then shows how the same Fisher hyperedges can be used as an interpretable inductive bias for message passing on correlator observables.
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