I'd like to apply the Geometric Product from Geometric Algebra as an activation function in Deep Learning. Could you provide an example of how I could implement the Geometric Product as a replacement to the dot product in neural networks?
The geometric product (also called "dot product") is a special case of the Clifford algebra, where it reduces to the usual dot product.

In general, there is no direct implementation of this operator for deep learning because the number of parameters required by such model would become too large and overparameterized, which can lead to instability during training. However, some approximations exist that reduce computational requirements while still preserving the main properties of the original operation [2]. One approach uses matrix multiplications based on SVD or other low rank approximation algorithms. Another method replaces the multiplication with a more efficient inner products using Gram matrices derived from the kernel functions used in the network, similar to what is done in convolutional layers.