Covariance-operator distance

A covariance-operator distance compares positive trace-class operators while respecting either their linear embedding in operator space or a chosen factorization geometry.

The linear distance between covariance operators is $d_L(C_1,C_2)=\Vert C_1-C_2{\Vert }_{\mathrm{HS}}$.

The square-root distance is $d_R(C_1,C_2)=\Vert C_1^{1/2}-C_2^{1/2}{\Vert }_{\mathrm{HS}}$.

For covariance operators $C_1,\dots ,C_n$, their Fréchet mean under the square-root distance isThis follows because taking positive square roots embeds the operators into the Hilbert space of self-adjoint Hilbert-Schmidt operators.

For factorizations $C_i=L_i{L}_i^{\ast }$, the Procrustes distance is $d_P(C_1,C_2)={\inf }_R\Vert L_1-L_{2}R{\Vert }_{\mathrm{HS}}$, where $R$ ranges over unitary operators on the factor space. Equivalently,