The weakest possible bilinear embedding that respects translation invariance is just a lookup table from the displacement to an arbitrary learnable square matrix of the proper dimension.

One reason algebraic structure is often nice is that it saves computation by reducing degrees of freedom. With so many extra parameters, in a naïve MacBook implementation, this unclever function took around 5 times longer to test than the previous ones.

Although it performs fine in a toy test, this function is obviously a bad way to spend FLOPs and would be very brittle in practice. (5/7)