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