Paths to Overcome Limitations Matrix Multiplication Bottleneck in Training

The core idea is that we may be using the wrong mathematical objects to represent information in transformers. Quaternions, the algebra behind 3D rotation, offer a structurally different forward pass. Replace real valued projections in attention with Hamilton products over quaternion valued inputs, and you get the same effect at a quarter of the parameter cost. The more interesting angle is geometric. Methods like LoRA work because fine-tuning updates cluster in a low dimensional rotational sub