From Information Geometry to Jet Substructure: A Triality of Cumulant Tensors, Energy Correlators, and Hypergraphs

2026-05-04

AI Summary

This work addresses the limitation of traditional pairwise Fisher information matrices in distinguishing irreducible multi-observable radiation patterns from ordinary pairwise correlations, which arises from their neglect of higher-order structural information. By leveraging local tensors in natural exponential family coordinates, the study establishes a tripartite equivalence among Kullback–Leibler expansion coefficients, connected cumulants, and hypergraph hyperedge weights. This framework introduces, for the first time, higher-order Fisher tensors into a physics-informed hypergraph representation. Through an exact correspondence between Fisher tensors, energy correlation functions, and hypergraphs, the method enables precise identification of multi-observable radiation structures and guides both observable basis compression and hypergraph learning. Evaluated on jet substructure classification, basis compression, and few-shot learning tasks, the approach demonstrates significant performance gains, with cubic Fisher tensors effectively reducing KL truncation error and enhancing ternary structure discrimination.

Abstract

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.

Problem

• jet substructure
• higher-order correlations
• cumulant tensors
• energy correlators
• hypergraphs

Innovation

• Fisher tensors
• energy correlators
• hypergraphs
• cumulants
• exponential-family manifolds