For positive trace-class covariance operators $C_i=L_iL_i^*$, their <Procrustes distance between covariance operators> is
$$
d_P(C_1,C_2)
=\inf_{R\in\mathcal O(H)}\lVert L_1-L_2R\rVert_{\mathrm{HS}},
$$
where $\mathcal O(H)$ is the group of <unitary operators> on the real Hilbert space $H=L^2[0,1]$. Expanding the square gives the equivalent formula
$$
d_P(C_1,C_2)^2
=\lVert L_1\rVert_{\mathrm{HS}}^2+\lVert L_2\rVert_{\mathrm{HS}}^2
-2\sup_{R\in\mathcal O(H)}\operatorname{tr}(R^*L_2^*L_1).
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 225 / 3 / b / i / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-225/3/b/i/solution