TY - JOUR
T1 - How Equivariance Reduces Sample Complexity in Materials GNNs: A Theoretical Bound
AU - Hiroshi Nakamura
AU - Yuta Kato
JF - Journal of Computational and Data-Driven Materials Engineering
JO - J. Comput. Data-Driven Mater. Eng.
SN - 3149-9368
Y1 - 2023
VL - 2
IS - 1
DO - 10.68159/f996793723
SP - 13
N2 - Equivariant graph neural networks have become central to data-efficient modeling of interatomic potentials, yet the mechanism underlying their empirical advantage remains insufficiently quantified. This work establishes a theoretical link between E(3)/SE(3) equivariance and sample complexity reduction in materials GNNs. By constraining the hypothesis space to functions that respect physical symmetries, equivariant architectures eliminate orbit-related redundancy and reduce the effective dimension of the learning problem. Within a PAC-style framework, we derive a bound showing that the sample complexity of an equivariant hypothesis class is reduced relative to its unrestricted counterpart by a factor proportional to the symmetry group size |G|, up to logarithmic terms. For discrete crystallographic groups, this yields finite reductions (e.g., 48× for cubic symmetry), while for continuous groups such as SO(3), the reduction becomes unbounded in the finite-data regime. The analysis relies on group-averaging projections and the induced contraction of VC dimension. These results provide a quantitative explanation for the observed data efficiency of architectures such as NequIP, MACE, and Allegro, and clarify the limitations of non-equivariant baselines such as SchNet. The framework situates symmetry as a statistically optimal inductive bias and offers principled guidance for designing data-efficient models in materials discovery.
UR - https://iamrp.net/f996793723
ER -