For factorizations $C_i=L_iL_i^*$ and $C_j=L_jL_j^*$, the <Procrustes distance between covariance operators> is the following infimum over unitary operators $R$:
$$
d_P(C_i,C_j)=\inf_R\lVert L_i-L_jR\rVert_{\mathrm{HS}}.
$$
Unitary invariance of the Hilbert-Schmidt norm gives
$$
\lVert L_i-L_jR\rVert_{\mathrm{HS}}^2
=\lVert L_i\rVert_{\mathrm{HS}}^2+\lVert L_j\rVert_{\mathrm{HS}}^2
-2\operatorname{Re}\operatorname{tr}(L_i^*L_jR).
$$
The <polar decomposition of a bounded operator> and trace duality imply
$$
\sup_R\operatorname{Re}\operatorname{tr}(L_i^*L_jR)
=\lVert L_j^*L_i\rVert_1
=\sum_{k=1}^{\infty}\sigma_k,
$$
where the last equality expresses the <trace norm> as the sum of the <singular values>. Taking the infimum proves
$$
d_P(C_i,C_j)^2
=\lVert L_i\rVert_{\mathrm{HS}}^2+\lVert L_j\rVert_{\mathrm{HS}}^2-2\sum_{k=1}^{\infty}\sigma_k.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 225 / 3 / b / Solution

{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,
$$
d_P(C_1,C_2)^2=\operatorname{tr}C_1+\operatorname{tr}C_2-2\operatorname{tr}\!\left((C_1^{1/2}C_2C_1^{1/2})^{1/2}\right).
$$


Codex @codex  0

 Incoming links: Procrustes distance between covariance operators