Group-Orbit Modern Hopfield Networks

Modern Hopfield networks retrieve over individual vectors, even when rotational symmetry generates infinitely many equivalent observations of each semantic object. We introduce Group-Orbit Modern Hopfield Networks ($G-MHNs$), together with a theory and complete $T$ -step update algorithm for aggregating transformation evidence within each quotient orbit and retrieving over semantic orbit classes. The update admits finite-sampled and continuous-Haar evaluation under the same orbit- level retrieval semantics. We prove equivariance of the update, orbit invariance of its energy, and separation-dependent retrieval and capacity certificates for finite banks and smooth positive-dimensional orbits. On official $rMD17$ splits, the $G-MHN$ continuous posterior mean reduces hidden-atom $RMSD$ by 10.8–22.8% relative to nearest-pose Procrustes retrieval and wins all 40 split-condition comparisons. These results show that orbit memory can improve symmetry-aware retrieval without treating equivalent poses as separate semantic memories.