Geometric Algebra Layers Excel in Compositional 3D Tasks

Researchers investigated the specific conditions under which neural networks built with Clifford algebra (Cl(3,0)) primitives, which are inherently SO(3)-equivariant, offer advantages over simpler scalarization approaches. Both methods are designed to learn 3D vector laws efficiently. The study compared their performance on various tasks, from simple single-stage operations to complex compositional targets. For straightforward tasks like rotation or cross products, scalarization, which uses invariant dot products fed into a small MLP, matched or even surpassed the geometric algebra networks with less training cost. However, a significant advantage for Cl(3,0) networks emerged in compositional tasks involving nested group operations, such as applying multiple rotations sequentially. In these low-data regimes, geometric algebra layers achieved with 100 samples what scalarization required 3000 samples for, a gap that persisted even with stronger baselines. The findings suggest that geometric algebra layers are not a universal shortcut but become particularly useful when the target function requires deep composition of group elements.