# Subspace Detours Meet Gromov-Wasserstein
## 3 Subspace Detours for \(Gw\)
### Properties  
Let \(E\subset\mathbb{R}^{p}\) and \(F\subset\mathbb{R}^{q}\) and denote:  
\[GW_{E,F}(\mu,\
u)=\inf_{\gamma\in\Pi_{E,F}(\mu,\
u)}\iint L(x,x^{\prime},y,y^{ \prime})\mathrm{d}\gamma(x,y)\mathrm{d}\gamma(x^{\prime},y^{\prime}) \tag{5}\]  
the Gromov-Wasserstein problem restricted to subspace optimal plans (2). In the following, we show that Monge-Knothe couplings are optimal plans of this problem, which is a direct transposition of Proposition 1 in [14]. **Proposition 1**. : _Let \(\mu\in\mathcal{P}(\mathbb{R}^{p})\) and \(v\in\mathcal{P}(\mathbb{R}^{q})\), \(E\subset\mathbb{R}^{p}\), \(F\subset\mathbb{R}^{q}\), \(\tau_{\mathrm{MK}}=\gamma^{*}_{E\times F}\otimes\gamma^{*}_{E^{\perp}\times F ^{\perp}|E\times F}\), where \(\gamma^{*}_{E\times F}\) is an optimal coupling between \(\mu_{E}\) and \(v_{F}\), and for \(\gamma^{*}_{E\times F}\), almost every \((x_{E},y_{F})\), \(\gamma^{*}_{E^{\perp}\times F^{\perp}|E\times F}\big{(}(x_{E},y_{F}),\cdot\big{)}\) is an optimal coupling between \(\mu_{E^{\perp}|E}(x_{E},\cdot)\) and \(v_{F^{\perp}|F}(y_{F},\cdot)\). Then we have:_  
\[\pi_{\mathrm{MK}}\in\operatorname*{argmin}_{\gamma\in\Pi_{E,F}(\mu,\
u)}\iint L (x,x^{\prime},y,y^{\prime})\mathrm{d}\gamma(x,y)\mathrm{d}\gamma(x^{\prime},y^ {\prime}).\]  
Proof. : Let \(\gamma\in\Pi_{E,F}(\mu,\
u)\), then:  
\[\iint L(x,x^{\prime},y,y^{\prime})\mathrm{d}\gamma(x,y)\mathrm{d} \gamma(x^{\prime},y^{\prime})\] \[=\iint\Big{(}\iint L(x,x^{\prime},y,y^{\prime})\gamma_{E^{\perp} \times F^{\perp}|E\times F}\big{(}(x_{E},y_{F}),(\mathrm{d}x_{E^{\perp}}, \mathrm{d}y_{F^{\perp}})\big{)}\] \[\gamma_{E^{\perp}\times F^{\perp}|E\times F}\big{(}(x_{E}^{\prime },y_{F}^{\prime}),(\mathrm{d}x_{E^{\perp}}^{\prime},\mathrm{d}y_{F^{\perp}}) \big{)}\Big{)}\mathrm{d}\gamma^{*}_{E\times F}(x_{E},y_{F})\mathrm{d}\gamma^{*} _{E\times F}(x_{E}^{\prime},y_{F}^{\prime}).\]  
However, for \(\gamma^{*}_{E\times F}\) a.e. \((x_{E},y_{F}),(x_{E}^{\prime},y_{F}^{\prime})\),  
\[\iint L(x,x^{\prime},y,y^{\prime})\gamma_{E^{\perp}\times F^{ \perp}|E\times F}\big{(}(x_{E},y_{F}),(\mathrm{d}x_{E^{\perp}},\mathrm{d}y_{ F^{\perp}})\big{)}\gamma_{E^{\perp}\times F^{\perp}|E\times F}\big{(}(x_{E}^{ \prime},y_{F}^{\prime}),(\mathrm{d}x_{E^{\perp}}^{\prime},\mathrm{d}y_{F^{ \perp}}^{\prime})\big{)}\] \[\geq\iint L(x,x^{\prime},y,y^{\prime})\gamma^{*}_{E^{\perp}\times F ^{\perp}|E\times F}\big{(}(x_{E},y_{F}),(\mathrm{d}x_{E^{\perp}},\mathrm{d}y_ {F^{\perp}})\big{)}\gamma^{*}_{E^{\perp}\times F^{\perp}|E\times F}\big{(}(x_{ E}^{\prime},y_{F}^{\prime}),(\mathrm{d}x_{E^{\perp}}^{\prime},\mathrm{d}y_{F^{ \perp}}^{\prime})\big{)}\]  
by definition of the Monge-Knothe coupling. By integrating with respect to \(\gamma^{*}_{E\times F}\), we obtain:  
\[\iint L(x,x^{\prime},y,y^{\prime})\mathrm{d}\gamma(x,y)\mathrm{d}\gamma(x^{ \prime},y^{\prime})\geq\iint L(x,x^{\prime},y,y^{\prime})\mathrm{d}\pi_{ \mathrm{MK}}(x,y)\mathrm{d}\pi_{\mathrm{MK}}(x^{\prime},y^{\prime}).\]  
Therefore, \(\pi_{\mathrm{MK}}\) is optimal for subspace optimal plans. The key properties of \(GW\) that we would like to keep are its invariances. We show in two particular cases that we conserve them on the orthogonal spaces (since the measure on \(E\times F\) is fixed). **Proposition 2**. : _Let \(\mu\in\mathcal{P}(\mathbb{R}^{p})\), \(v\in\mathcal{P}(\mathbb{R}^{q})\), \(E\subset\mathbb{R}^{p}\), \(F\subset\mathbb{R}^{q}\)._  
_For \(L(x,x^{\prime},y,y^{\prime})=\big{(}\|x-x^{\prime}\|_{2}^{2}-\|y-y^{\prime}\|_ {2}^{2}\big{)}^{2}\) or \(L(x,x^{\prime},y,y^{\prime})=\big{(}\langle x,x^{\prime}\rangle_{p}-\langle y,y^{\prime}\rangle_{q}\big{)}^{2}\), \(GW_{E,F}\) (5) is invariant with respect to isometries of the form \(f=(Id_{E},f_{E^{\perp}})\) (resp. \(g=(Id_{F},g_{E^{\perp}})\)) with \(f_{E^{\perp}}\) an isometry on \(E^{\perp}\) (resp. \(S_{F^{\perp}}\) an isometry on \(F^{\perp}\)) with respect to the corresponding cost \((c(x,x^{\prime})=\|x-x^{\prime}\|_{2}^{2}\) or \(c(x,x^{\prime})=\langle x,x^{\prime}\rangle_{p}\))._  
Proof. : We propose a sketch of the proof.