# Subspace Detours Meet Gromov-Wasserstein
## Appendix A Subspace Detours
### Closed-Form between Gaussians
#### a.2.1 Quadratic GW Problem
First:  
\[T_{E,F}\Sigma_{E}C^{T}+T_{E,F}\Sigma_{EE^{\perp}}T_{E^{\perp},F^ {\perp}}^{T} =T_{E,F}\Sigma_{E}\Sigma_{E}^{-1}T_{E,F}^{-1}\Lambda_{F^{\perp}F} ^{T}-T_{E,F}\Sigma_{E}\Sigma_{E}^{-1}\Sigma_{E^{\perp}E}^{T}T_{E^{\perp},F^{ \perp}}^{T}+T_{E,F}\Sigma_{EE^{\perp}}T_{E^{\perp},F^{\perp}}^{T}\] \[=\Lambda_{FF^{\perp}}.\]  
For the last equation:  
\[(C\Sigma_{E}+T_{E^{\perp},F^{\perp}}\Sigma_{E^{\perp}E})C^{T}+(C \Sigma_{EE^{\perp}}+T_{E^{\perp},F^{\perp}}\Sigma_{E^{\perp}})T_{E^{\perp}F^{ \perp}}^{T}\] \[=(\Lambda_{F^{\perp}F}(T_{E,F}^{T})^{-1}-T_{E^{\perp},F^{\perp}} \Sigma_{E^{\perp}E}+T_{E^{\perp},F^{\perp}}\Sigma_{E^{\perp}E})\Sigma_{E}^{-1 }(T_{E,F}^{-1}\Lambda_{F^{\perp}F^{\perp}}^{T}-\Sigma_{E^{\perp}E}^{-1}T_{E^{ \perp}E^{\perp}}^{T}T_{E^{\perp},F^{\perp}}^{T})\] \[+\Lambda_{F^{\perp}F}(T_{E,F}^{T})^{-1}\Sigma_{E}^{-1}\Sigma_{E ^{\perp}E}.T_{E^{\perp}F^{\perp}}^{T}-T_{E^{\perp},F^{\perp}}\Sigma_{E^{ \perp}E}\Sigma_{E}^{-1}\Sigma_{E^{\perp}E}\Sigma_{E^{\perp}}^{T}T_{E^{\perp},F^{\perp}}^{T}+T_{E^{\perp},F^{\perp}}\Sigma_{E^{\perp}}T_{E^{\perp},F^{ \perp}}^{T}\] \[=\Lambda_{F^{\perp}F}(T_{E,F}^{T})^{-1}\Sigma_{E}^{-1}T_{E,F}^{-1 }\Lambda_{F^{\perp}F}^{T}-\Lambda_{F^{\perp}F}(T_{E,F}^{T})^{-1}\Sigma_{E}^{- 1}\Sigma_{E^{\perp}E}^{T}T_{E^{\perp},F^{\perp}}^{T}-T_{E^{\perp},F^{\perp}} \Sigma_{E^{\perp}E}\Sigma_{E}^{-1}T_{E,F}^{-1}\Lambda_{F^{\perp}F}^{T}\] \[+T_{E^{\perp},F^{\perp}}\Sigma_{E}\Sigma_{E}^{-1}\Sigma_{E}^{T} T_{E^{\perp},F^{\perp}}^{T}+T_{E^{\perp},F^{\perp}}\Sigma_{E^{\perp}E}^{-1}T_{E ^{\perp}F}^{T}\Lambda_{F^{\perp}F}^{T}-T_{E^{\perp},F^{\perp}}\Sigma_{E^{ \perp}E}^{-1}\Sigma_{E}^{-1}\Sigma_{E^{\perp}E}^{-1}\Sigma_{E^{\perp}E}^{T} T_{E^{\perp}F^{\perp}}^{T}\] \[+\Lambda_{F^{\perp}F}(T_{E,F}^{T})^{-1}\Sigma_{E}^{-1}\Sigma_{E ^{\perp}E}.T_{E^{\perp},F^{\perp}}^{T}-T_{E^{\perp},F^{\perp}}\Sigma_{E^{ \perp}E}\Sigma_{E}^{-1}\Sigma_{E^{\perp}E}^{T}T_{E^{\perp},F^{\perp}}^{T}+T_ {E^{\perp},F^{\perp}}\Sigma_{E^{\perp}}T_{E^{\perp},F^{\perp}}^{T}\] \[C=(\Lambda_{F^{\perp}F}(T_{E,F}^{T})^{-1}-T_{E^{\perp},F^{\perp}}\Sigma_{E^{ \perp}E})\Sigma_{E}^{-1}.\]0  
Now, using that \((T_{E,F}^{T})^{-1}\Sigma_{E}^{-1}T_{E,F}^{-1}=(T_{E,F}\Sigma_{E}T_{E,F}^{T})^{-1}= \Lambda_{F}^{-1}\) and \(\Sigma_{E^{\perp}}-\Sigma_{E^{\perp}E}\Sigma_{E}^{-1}\Sigma_{E^{\perp}E}^{T}= \Sigma/\Sigma_{E}\), we have:  
\[C=(\Lambda_{F^{\perp}F}(T_{E,F}^{T})^{-1}-T_{E^{\perp},F^{\perp}}\Sigma_{E^{ \perp}E})\Sigma_{E}^{-1}.\]1 \[C=(\Lambda_{F^{\perp}F}(T_{E,F}^{T})^{-1}-T_{E^{\perp},F^{\perp}}\Sigma_{E^{ \perp}E})\Sigma_{E}^{-1}.\]2 \[C=(\Lambda_{F^{\perp}F}(T_{E,F}^{T})^{-1}-T_{E^{\perp},F^{\perp}}\Sigma_{E^{ \perp}E})\Sigma_{E}^{-1}.\]3 \[C=(\Lambda_{F^{\perp}F}(T_{E,F}^{T})^{-1}-T_{E^{\perp},F^{\perp}}\Sigma_{E^{ \perp}E})\Sigma_{E}^{-1}.\]4  
Then, \(\pi_{\text{MK}}\) is of the form \((Id,T_{\text{MK}})_{\#}\mu\) with:  
\[C=(\Lambda_{F^{\perp}F}(T_{E,F}^{T})^{-1}-T_{E^{\perp},F^{\perp}}\Sigma_{E^{ \perp}E})\Sigma_{E}^{-1}.\]5  
Appendix A.2.2. Closed-Form between Gaussians for Monge-Independent  
Suppose is \(k\geq k^{\prime}\) in order to be able to define the OT map between \(\mu_{E}\) and \(v_{F}\). For the Monge-Independent plan, \(\pi_{\text{MI}}=\gamma_{E\times F}^{*}\otimes(\mu_{E^{\perp}|E}\otimes v_{F^{\perp} |F})\), let \((X,Y)\sim\pi_{\text{MI}}\). We know that \(\pi_{\text{MI}}\) is a degenerate Gaussian with a covariance of the form:  
\[\text{Cov}(X,Y)= \begin{pmatrix}\text{Cov}(X)&C\\ C^{T}&\text{Cov}(Y)\end{pmatrix} \]  
where \(\text{Cov}(X)=\Sigma\) and \(\text{Cov}(Y)=\Lambda\). Moreover, we know that \(C\) is of the form:  
\[ \begin{pmatrix}\text{Cov}(X_{E},Y_{F})&\text{Cov}(X_{E},Y_{F^{\perp}})\\ \text{Cov}(X_{E^{\perp}},Y_{F})&\text{Cov}(X_{E^{\perp}},Y_{F^{\perp}})\end{pmatrix}