## Quasiorthonormal Encoding

Our example with the Japanese car makes:

Make One-Hot Quasiorthonornal Code
Toyota (1,0,0,0,0) (0.851, 0.000, -0.526)
Honda (0,1,0,0,0) (0.526, -0.851, 0.000)
Subaru (0,0,1,0,0) (0.000, -0.526, 0.851)
Nissan (0,0,0,1,0) (0.851, 0.000, 0.526)
Mitsubishi (0,0,0,0,1) (-0.526, -0.851, 0.000)

If you think of one-hot encoding as encoding categories with unit vectors in $N$ dimensions ($\mathbb{R}^5$ in this case), then you are encoding your categories onto an orthonormal basis (the set of unit vectors in $\mathbb{R}^N$. Quasiorthonormal encoding relaxes the orthonormal requirement and substitute a quasiorthonormal "basis". The above table shows 5 quasiorthonormal vectors in 3 dimensions having a minimum mutual angle of aboout $63^\circ$.

There is substantial mathematical theory on what is quasiorthogonality and quasiorthonormality, how you construct it, how to decode it including a counterpart to the softmax function. All of which is presented in much greater detail in the accompanying proceedings paper.

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