# Subspace Detours Meet Gromov-Wasserstein
## Appendix B Knothe-Rosenblatt
### Proof of Theorem 3  
We first recall a useful theorem. **Theorem A1** (Theorem 2.8 in Billingsley [45]). : _Let \(\Omega=X\times Y\) be a separable space, and let \(P,P_{n}\in\mathcal{P}(\Omega)\) with marginals \(P_{X}\) (respectively \(P_{n,X}\)) and \(P_{Y}\) (respectively \(P_{n,Y}\)). Then, \(P_{n,X}\otimes P_{n,Y}\xrightarrow{\mathcal{D}}P\) if and only if \(P_{n,X}\xrightarrow{\mathcal{D}}P_{X}\), \(P_{n,Y}\xrightarrow{\mathcal{D}}P_{Y}\) and \(P=P_{X}\otimes P_{Y}\)._  
Proof of Theorem 3.: The following proof is mainly inspired by the proof of Theorem 1 in [21] (Theorem 2.1), [22] (Theorem 3.1.6) and [19] (Theorem 2.23). Let \(\mu,\
u\in\mathcal{P}(\mathbb{R}^{d})\), absolutely continuous, with finite fourth moments and compact supports. We recall the problem \(\mathcal{H}\mathcal{W}_{t}\):  
\[\mathcal{H}\mathcal{W}_{t}^{2}(\mu,\
u)=\inf_{\gamma\in\Pi(\mu,\
u)}\iint \sum_{k=1}^{d}\Big{(}\prod_{i=1}^{k-1}\lambda_{t}^{(i)}\Big{)}\;(x_{k}x_{k}^ {\prime}-y_{k}y_{k}^{\prime})^{2}\;\mathrm{d}\gamma_{t}(x,y)\mathrm{d}\gamma_ {t}(x^{\prime},y^{\prime}),\]  
with \(\forall t>0,\;\forall i\in\{1,\ldots,d-1\}\), \(\lambda_{t}^{(i)}>0\) and \(\lambda_{t}^{(i)}\xrightarrow[t\to 0]{}0\). First, let us denote \(\gamma_{t}\) the optimal coupling for \(\mathcal{H}\mathcal{W}_{t}\) for all \(t>0\). We want to show that \(\gamma_{t}\xrightarrow[t\to 0]{}\gamma_{K}\) with \(\gamma_{K}=(Id\times T_{K})_{\mathrm{z}}\mu\) and \(T_{K}\) our alternate Knothe-Rosenblatt rearrangement. Let \(\gamma\in\Pi(\mu,\
u)\) such that \(\gamma_{t}\xrightarrow[t\to 0]{}\gamma\) (true up to subsequence as \(\{\mu\}\) and \(\{\
u\}\) are tight in \(\mathcal{P}(X)\) and \(\mathcal{P}(Y)\) if \(X\) and \(Y\) are polish space, therefore, by [18] (Lemma 4.4), \(\Pi(\mu,\
u)\) is a tight set, and we can apply the Prokhorov theorem [19] (Box 1.4) on \((\gamma_{t})_{t}\) and extract a subsequence). _Part 1_:  
First, let us notice that:  
\[\mathcal{H}\mathcal{W}_{t}^{2}(\mu,\
u) =\iint\sum_{k=1}^{d}\Big{(}\prod_{i=1}^{k-1}\lambda_{t}^{(i)} \Big{)}\;(x_{k}x_{k}^{\prime}-y_{k}y_{k}^{\prime})^{2}\;\mathrm{d}\gamma_{t}(x,y)\mathrm{d}\gamma_{t}(x^{\prime},y^{\prime})\] \[=\iint(x_{1}x_{1}^{\prime}-y_{1}y_{1}^{\prime})^{2}\;\mathrm{d} \gamma_{t}(x,y)\mathrm{d}\gamma_{t}(x^{\prime},y^{\prime})\] \[+\iint\sum_{k=2}^{d}\Big{(}\prod_{i=1}^{k-1}\lambda_{t}^{(i)} \Big{)}\;(x_{k}x_{k}^{\prime}-y_{k}y_{k}^{\prime})^{2}\;\mathrm{d}\gamma_{t}(x,y)\mathrm{d}\gamma_{t}(x^{\prime},y^{\prime}).\]Moreover, as \(\gamma_{t}\) is the optimal coupling between \(\mu\) and \(\
u\), and \(\gamma_{K}\in\Pi(\mu,\
u)\),  
\[\mathcal{H}\mathcal{W}_{t}^{2}(\mu,\
u) \leq\iint\sum_{k=1}^{d}\left(\prod_{i=1}^{k-1}\lambda_{t}^{(i)} \right)\,(x_{k}x_{k}^{\prime}-y_{k}y_{k}^{\prime})^{2}\,\mathrm{d}\gamma_{K}(x,y)\mathrm{d}\gamma_{K}(x^{\prime},y^{\prime})\] \[=\iint(x_{1}x_{1}^{\prime}-y_{1}y_{1}^{\prime})^{2}\,\mathrm{d} \gamma_{K}(x,y)\mathrm{d}\gamma_{K}(x^{\prime},y^{\prime})\] \[+\iint\sum_{k=2}^{d}\left(\prod_{i=1}^{k-1}\lambda_{t}^{(i)} \right)\,(x_{k}x_{k}^{\prime}-y_{k}y_{k}^{\prime})^{2}\,\mathrm{d}\gamma_{K}(x,y)\mathrm{d}\gamma_{K}(x^{\prime},y^{\prime}).\]  
In our case, we have \(\gamma_{t}\xrightarrow[t\to 0]{\mathcal{D}}\gamma\), thus, by Theorem A1, we have \(\gamma_{t}\otimes\gamma_{t}\xrightarrow[t\to 0]{\mathcal{D}}\gamma\otimes\gamma\). Using the fact that \(\forall i,\ \lambda_{t}^{(i)}\xrightarrow[t\to 0]{\mathcal{D}}0\) (and Lemma 1.8 of Santambrogio [19], since we are on compact support, we can bound the cost (which is continuous) by its max), we obtain the following inequality  
\[\iint(x_{1}x_{1}^{\prime}-y_{1}y_{1}^{\prime})^{2}\,\mathrm{d}\gamma(x,y) \mathrm{d}\gamma(x^{\prime},y^{\prime})\leq\iint(x_{1}x_{1}^{\prime}-y_{1}y_{ 1}^{\prime})^{2}\,\mathrm{d}\gamma_{K}(x,y)\mathrm{d}\gamma_{K}(x^{\prime},y^ {\prime}).\]  
By denoting \(\gamma^{1}\) and \(\gamma_{K}^{1}\) the marginals on the first variables, we can use the projection \(\pi^{1}(x,y)=(x_{1},y_{1})\), such as \(\gamma^{1}=\pi_{\#}^{1}\gamma\) and \(\gamma_{K}^{1}=\pi_{\#}^{1}\gamma_{K}\).