Speaker
Description
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 the higher Fisher tensors supply this missing structure. 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: it is 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 measured or simulated correlator data. Extending the quadratic Fisher matrix to the first non-trivial higher tensor identifies connected multi-observable radiation patterns and supplies hyperedge weights for higher-order Laplacians and message passing. It also gives a criterion for compressing observable bases beyond pairwise information. We develop these constructions and explain why the exact cumulant interpretation is specific to natural exponential-family coordinates.
We illustrate the framework in four applications. In a minimal local-KL study, including the cubic Fisher tensor reduces the KL truncation error by about 30× near the reference point and isolates the dominant triplet structure. In a W → q<span style="text-decoration: overline;">q</span> versus t → bq<span style="text-decoration: overline;">q</span> substructure benchmark, the hypergraph selector improves compressed-basis classification. The Fisher hypergraph retains more third-order local response at twelve observables, with <span style="text-decoration: overline;">R</span>(3) = 0.937 compared with 0.871 for the pairwise graph. A low-capacity learning benchmark then shows that the same Fisher hyperedges serve as an interpretable inductive bias for message passing over correlator observables.