# Subspace Detours Meet Gromov-Wasserstein
## 4 Triangular Coupling as Limit of Optimal Transport Plans for Quadratic Cost
### Properties  
First, we derive some useful properties of (12) which are usual for the regular Gromov-Wasserstein problem. Formally, we show that the problem is well defined and that it is a pseudometric with invariances with respect to axes. **Proposition 6**. : _Let \(\mu,\
u\in\mathcal{P}(\mathbb{R}^{d})\)._  
1. _The problem (_12_) always admits a minimizer._
2. \(\mathcal{H}\mathcal{W}\) _is a pseudometric (i.e., it is symmetric, non-negative,_ \(\mathcal{H}\mathcal{W}(\mu,\mu)=0\)_, and it satisfies the triangle inequality)._
3. \(\mathcal{H}\mathcal{W}\) _is invariant to reflection with respect to axes._  
Proof. : Let \(\mu,\
u\in\mathcal{P}(\mathbb{R}^{d})\),  
1. \((x,x^{\prime})\mapsto x\odot x^{\prime}\) is a continuous map, therefore, \(L\) is less semi-continuous. Hence, by applying Lemma 2.2.1 of [26], we observe that \(\gamma\mapsto\iint L(x,x^{\prime},y,y^{\prime})\mathrm{d}\gamma(x,y)\mathrm{d} \gamma(x^{\prime},y^{\prime})\) is less semi-continuous for the weak convergence of measures. Now, as \(\Pi(\mu,\
u)\) is a compact set (see the proof of Theorem 1.7 in [19] for the Polish space case and of Theorem 1.4 for the compact metric space) and \(\gamma\mapsto\iint L\mathrm{d}\gamma\mathrm{d}\gamma\) isless semi-continuous for the weak convergence, we can apply the Weierstrass theorem (Memo 2.2.1 in [26]), which states that (12) always admits a minimizer. 2. See Theorem 16 in [25]. 3. For invariances, we first look at the properties that must be satisfied by \(T\) in order to have: \(\forall x,x^{\prime},\ f(x,x^{\prime})=f(T(x),T(x^{\prime}))\) where \(f:(x,x^{\prime})\mapsto x\odot x^{\prime}\). We find that \(\forall x\in\mathbb{R}^{d},\ \forall 1\leq i\leq d,\ ||T(x)|_{i}|=|x_{i}|\) because, denoting \((e_{i})_{i=1}^{d}\) as the canonical basis, we have: \[x\odot e_{i}=T(x)\odot T(e_{i}),\] which implies that: \[x_{i}=[T(x)]_{i}[T(e_{i})]_{i}.\] However, \(f(e_{i},e_{i})=f(T(e_{i}),T(e_{i}))\) implies \([T(e_{i})]_{i}^{2}=1\), and therefore: \[|[T(x)]_{i}|=|x_{i}|.\] If we take for \(T\) the reflection with respect to axis, then it satisfies \(f(x,x^{\prime})=f(T(x),T(x^{\prime}))\) well. Moreover, it is a good equivalence relation, and therefore, we have a distance on the quotient space. \(\mathcal{HW}\) loses some properties compared to \(GW\). Indeed, it is only invariant with respect to axes, and it can only compare measures lying in the same Euclidean space in order for the distance to be well defined. Nonetheless, we show in the following that we can derive some links with triangular couplings in the same way as the Wasserstein distance with KR. Indeed, the cost \(L\) (13) is separable and reduces to the inner-GW loss in 1D, for which we have a closed-form solution. We can therefore define a degenerated version of it:  
\[ \begin{split}\forall x,x^{\prime},y,y^{\prime}\in\mathbb{R}^{d}, \ L_{t}(x,x^{\prime},y,y^{\prime})&=\sum_{k=1}^{d}\Big{(}\prod _{i=1}^{k-1}\lambda_{t}^{(i)}\Big{)}(x_{k}x^{\prime}_{k}-y_{k}y^{\prime}_{k}) ^{2}\\ &=(x\odot x^{\prime}-y\odot y^{\prime})^{T}A_{t}(x\odot x^{\prime }-y\odot y^{\prime})\end{split}  \tag{14}\]  
with \(A_{t}=\text{diag}(1,\lambda_{t}^{(1)},\lambda_{t}^{(1)}\lambda_{t}^{(2)}, \ldots,\prod_{i=1}^{d-1}\lambda_{t}^{(i)}),\) such as for all \(t>0\), and for all \(i\in\{1,\ldots,d-1\}\), \(\lambda_{t}^{(i)}>0\), and \(\lambda_{t}^{(i)}\xrightarrow[t\to 0]{}0\). We denote \(\mathcal{HW}_{t}\) the problem (12) with the degenerated cost (14). Therefore, we will be able to decompose the objective as:  
\[ \begin{split}&\iint L_{t}(x,x^{\prime},y,y^{\prime})\text{d} \gamma(x,y)\text{d}\gamma(x^{\prime},y^{\prime})\\ &=\iint(x_{1}x^{\prime}_{1}-y_{1}y^{\prime}_{1})^{2}\,\text{d} \gamma(x,y)\text{d}\gamma(x^{\prime},y^{\prime})\\ &+\iint\sum_{k=2}^{d}\left(\prod_{i=1}^{k-1}\lambda_{t}^{(i)} \right)(x_{k}x^{\prime}_{k}-y_{k}y^{\prime}_{k})^{2}\,\text{d}\gamma(x,y)\text {d}\gamma(x^{\prime},y^{\prime})\end{split} \]  
and to use the same induction reasoning as [21].