# Subspace Detours Meet Gromov-Wasserstein
## 3 Subspace Detours for \(Gw\)
### Closed-Form between Gaussians  
We can also derive explicit formulas between Gaussians in particular cases. Let \(q\leq p\), \(\mu=\mathcal{N}(m_{\mu},\Sigma)\in\mathcal{P}(\mathbb{R}^{p})\), \(\
u=\mathcal{N}(m_{\
u},\Lambda)\in\mathcal{P}(\mathbb{R}^{q})\) two Gaussian measures with \(\Sigma=P_{\mu}D_{\mu}P_{\mu}^{T}\) and \(\Lambda=P_{\
u}D_{\
u}P_{\
u}^{T}\). As previously, let \(E\subset\mathbb{R}^{p}\) and \(F\subset\mathbb{R}^{q}\) be \(k\) and \(k^{\prime}\) dimensional subspaces, respectively. Following [PERSON] and [PERSON] [14], we represent \(\Sigma\) in an orthonormal basis of \(E\oplus E^{\perp}\) and \(\Lambda\) in an orthonormal basis of \(F\oplus F^{\perp}\), i.e., \(\Sigma=\begin{pmatrix}\Sigma_{E}&\Sigma_{E^{\perp}}\\ \Sigma_{E^{\perp}E}&\Sigma_{E^{\perp}}\end{pmatrix}\). Now, let us denote the following:  
\[\Sigma/\Sigma_{E}=\Sigma_{E^{\perp}}-\Sigma_{E^{\perp}}^{T}\Sigma_{E^{\perp}}^ {-1}\]  
as the Schur complement of \(\Sigma\) with respect to \(\Sigma_{E}\). We know that the conditionals of Gaussians are Gaussians and that their covariances are the Schur complements (see, e.g., [28; 29]).  
For \(L(x,x^{\prime},y,y^{\prime})=\big{(}\|x-x^{\prime}\|_{2}^{2}-\|y-y^{\prime}\| _{2}^{2}\big{)}^{2}\), we have for now no certainty that the optimal transport plan is Gaussian. Let \(\mathcal{N}_{p+q}\) denote the set of Gaussians in \(\mathbb{R}^{p+q}\). By restricting the minimization problem to Gaussian couplings, i.e., by solving:  
\[\mathrm{GGW}(\mu,\
u)=\inf_{\gamma\in\Pi(\mu,\
u)\cap\mathcal{N}_{p+q}}\iint \big{(}\|x-x^{\prime}\|_{2}^{2}-\|y-y^{\prime}\|_{2}^{2}\big{)}^{2}\,\mathrm{ d}\gamma(x,y)\mathrm{d}\gamma(x^{\prime},y^{\prime}), \tag{6}\]  
[PERSON] et al. [30] showed that there is a solution \(\gamma^{*}=(Id,T)_{\#}\mu\in\Pi(\mu,\
u)\) with \(\mu=\mathcal{N}(m_{\mu},\Sigma)\), \(\
u=\mathcal{N}(m_{\
u},\Lambda)\) and  
\[\forall x\in\mathbb{R}^{d},\,T(x)=m_{\
u}+P_{\
u}AP_{\mu}^{T}(x-m_{\mu}) \tag{7}\]  
where \(A=\Big{(}I_{q}D_{\
u}^{\frac{1}{q}}(D_{\mu}^{(q)})^{-\frac{1}{2}}\quad 0_{q,p-q} \Big{)}\in\mathbb{R}^{q\times p}\), and \(\bar{I}_{q}\) is of the form \(\mathrm{diag}\big{(}(\pm 1)_{i\leq q}\big{)}\).  
By combining the results of [PERSON] and [PERSON] [14] and [PERSON] et al. [30], we obtain the following closed-form for Monge-Knothe couplings:  
**Proposition 3**.: _Suppose \(p\geq q\) and \(k=k^{\prime}\). For the Gaussian-restricted GW problem (6), a Monge-Knothe transport map between \(\mu=\mathcal{N}(m_{\mu},\Sigma)\in\mathcal{P}(\mathbb{R}^{p})\) and \(\
u=\mathcal{N}(m_{\
u},\Lambda)\in\mathcal{P}(\mathbb{R}^{q})\) is, for all \(x\in\mathbb{R}^{p}\), \(T_{\mathrm{MK}}(x)=m_{\
u}+B(x-m_{\mu})\) where:_  
\[B=\begin{pmatrix}T_{E,F}&0\\ C&T_{E^{\perp},F^{\perp}|E,F}\end{pmatrix}\]  
_with \(T_{E,F}\) being an optimal transport map between \(\mathcal{N}(0_{E},\Sigma_{E})\) and \(\mathcal{N}(0_{F},\Lambda_{F})\) (of the form (7)), \(T_{E^{\perp},F^{\perp}|E,F}\) an optimal transport map between \(\mathcal{N}(0_{E^{\perp}},\Sigma/\Sigma_{E})\) and \(\mathcal{N}(0_{F^{\perp}},\Lambda/\Lambda_{F})\), and \(\mathcal{C}\) satisfies:_  
\[C=(\Lambda_{F^{\perp}F}(T_{E,F}^{T})^{-1}-T_{E^{\perp},F^{\perp}|E,F}\Sigma_{E^ {\perp}E})\Sigma_{E}^{-1}.\]  
Proof.: See Appendix A.2.1.  
Suppose that \(k\geq k^{\prime}\), \(m_{\mu}=0\), and \(m_{\
u}=0\) and let \(T_{E,F}\) be an optimal transport map between \(\mu_{E}\) and \(v_{F}\) (of the form (7)). We can derive a formula for the Monge-Independent coupling for the inner-GW problem and the Gaussian restricted GW problem.  
Proposition 4: \(\pi_{\rm MI}=\mathcal{N}(0_{p+q},\Gamma)\) _where \(\Gamma=\begin{pmatrix}\Sigma&C\\ C^{T}&\Lambda\end{pmatrix}\) with_  
\[C=(V_{E}\Sigma_{E}+V_{E^{\perp}}\Sigma_{E^{\perp}E})T_{E,F}^{T}(V_{F}^{T}+ \Lambda_{F}^{-1}\Lambda_{F^{\perp}E}^{T}V_{F^{\perp}}^{T})\]  
_where \(T_{E,F}\) is an optimal transport map, either for the inner-GW problem or the Gaussian restricted problem._  
Proof: See Appendix A.2.2.