Procrustes distance between covariance operators
= Procrustes distance between covariance operators
{c}
{title2=$d_P$}
For factorizations $C_i=L_iL_i^*$, the Procrustes distance is $d_P(C_1,C_2)=\inf_R\lVert L_1-L_2R\rVert_{\mathrm{HS}}$, where $R$ ranges over unitary operators on the factor space. Equivalently,
$$
d_P(C_1,C_2)^2=\operatorname{tr}C_1+\operatorname{tr}C_2-2\operatorname{tr}\!\left((C_1^{1/2}C_2C_1^{1/2})^{1/2}\right).
$$