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

arXiv: https://arxiv.org/abs/2605.03063 · HTML: https://mlsanity.com/paper/2605.03063/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 3 of its 44 references that are in the map.
- First posted on arXiv 4 May 2026.

## Builds on (most cited first, 3 of 3)

- [Hypergraph Neural Networks](https://mlsanity.com/paper/1809.09401/) (2018)
- [ParticleNet: Jet Tagging via Particle Clouds](https://mlsanity.com/paper/1902.08570/) (2019)
- [Reconstructing short-lived particles using hypergraph representation learning](https://mlsanity.com/paper/2402.10149/) (2024)

## Built on by (most cited first, 1 of 1)

- [Probing SMEFT Operators through tbarttbart Production with Hyper-Graph Neural Networks at the LHC](https://mlsanity.com/paper/2605.18382/) (2026)

## Abstract (arXiv)

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.