Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 225 / 3

For positive trace-class operators, the square-root distance between covariance operators isIf $C_i=L_i{L}_i^{\ast }$ are Hilbert-Schmidt factorizations, the Procrustes distance between covariance operators iswhere $\mathcal{O}(H)$ is the group of orthogonal operators on the real separable Hilbert space $H$.

For an orthogonal $R$, unitary invariance of the Hilbert-Schmidt norm givesThe product $L_2^{\ast }{L}_1$ is trace-class. Its polar decomposition of a bounded operator and trace duality givewhere $({\sigma }_k)$ are its singular values. Taking the infimum proves

Hilbert-Schmidt convergence impliesThe Schatten norm Hölder inequality givesThe sum of singular values is the trace norm, and the reverse triangle inequality givesSubstitution into part b proves

Choose a unit vector $e$ and distinct positive numbers $a,b$, and setThese are distinct rank-one covariance operators. Their positive square roots are $\sqrt{a}(e\otimes e)$ and $\sqrt{b}(e\otimes e)$, soUsing these square roots as the factors in the Procrustes definition, the identity alignment attains the same value. The singular-value formula gives the matching lower bound,Hence $d_P=d_R$ although $C_1\ne C_2$.