Attractor Dynamics of Energy Transformer

Energy Transformer (ET) performs image completion by energy descent over image tokens, but its energy does not explicitly respect the translational structure of the image grid, limiting its ability to form continuous attractors. We introduce the Attractor Energy Transformer (AET), which incorporates a translation-invariant spatial kernel $\mathcal{K}$ on the toroidal token grid ${\mathbb{T}}^2$. We prove that the resulting spatial energy $F_{\mathrm{sp}}$ is invariant to cyclic translations, giving rise to a two-dimensional critical manifold of equivalent states. We further show that a rectangular boundary acts as a pinning perturbation that induces a restoring force along this manifold. On COCO2014 image completion, AET consistently outperforms ET, with the improvement persisting when the kernel topology is fixed. A sliding-window experiment provides further evidence for the theoretical analysis. The gap between the toroidal and rectangular topologies is smallest for an interior mask and larger at every offset that puts the masked region on the identified edge, consistent with the pinning predicted for the rectangle.