Putting aside the Quaterion unit basis [1, i, j, k] , W ⊗ Q can be expressed as: where We highlight that, there are only 4 distinct parameter variable elements (4 degrees of freedom), namely W r , W x , W y , W z , in the weight matrix (left) of (3), as illustrated by Figure 1; while in real-space feed-forward, all the elements of the weight matrix are different parameter variables (4 × 4 = 16 degrees of freedom).
the degrees of freedom in Quaternion feedforward is only a quarter of those in its real-space are used in 16 pairwise connections between components of the input and output Quaternions.