Congruence Coefficient#
\[\begin{split}
\begin{aligned}
\xi = r_c &= \frac{\sum_{ij}x_{ij}y_{ij}}{\sqrt{\left( \sum_{ij}x_{ij}^2 \right)\left( \sum_{ij}y_{ij}^2 \right)}} \\
&= \frac{\text{Tr}(\mathbf{XY^\top})}{\sqrt{\text{Tr}(\mathbf{XX^\top})\text{Tr}(\mathbf{YY^\top})}} \\
\end{aligned}
\end{split}\]
RV Coefficient#
Numerator -> The scalar produce between two PSD kernel matrices
Denominator -> Frobenius, Schur, Hilbert-Schmidt matrix scalar produce (Horn & Johnson, 1985, p261)
Cosine distance between two matrices