# Subspace Detours Meet Gromov-Wasserstein
## 5 Discussion  
We proposed in this work to extend the subspace detour approach to different subspaces, and to other optimal transport costs such as Gromov-Wasserstein. Being able to project on different subspaces can be useful when the data are not aligned and do not share the same axes of interest, as well as when we are working between different metric spaces as it is the case, for example, with graphs. However, a question that arises is how to choose these subspaces. Since the method is mostly interesting when we choose one-dimensional subspaces, we proposed to use a PCA and to project on the first directions for data embedded in Euclidean spaces. For more complicated data such as graphs, we projected onto the Fiedler vector and obtained good results in an efficient way on a 3D mesh registration problem. More generally, [PERSON] and [PERSON] [14] proposed to perform a gradient descent on the loss with respect to orthonormal matrices. This approach is non-convex and is only guaranteed to converge to a local minimum. Designing such an algorithm, which would minimize alternatively between two transformations in the Stiefel manifold, is left for future works. The subspace detour approach for transport problem is meaningful whenever one can identify subspaces that gather most of the information from the original distributions, while making the estimate more robust and with a better sample complexity as far as dimensions are lower. On the computational complexity side, and when we have only access to discrete data, the subspace detour approach brings better computational complexity solely when the subspaces are chosen as one dimensional. Indeed, otherwise, we have the same complexity for solving the subspace detour and solving the OT problem directly (since the complexity only depends on the number of samples). In this case, the 1D projection often gives distinct values for all the samples (for continuous valued data) and hence the Monge-Knothe coupling is exactly the coupling in 1D. As such, information is lost on  
Figure 3: Degenerated coupling. On the **first row**, the points are projected on their first coordinate and we plot the optimal coupling. On the **second row**, we plot the optimal coupling between the original points. the orthogonal spaces. It can be artificially recovered by quantizing the 1D values (as experimented in practice in [14]), but the added value is not clear and deserves broader studies. If given absolutely continuous distributions \(wt\). the Lebesgue measure, however, this limit does not exist but comes with the extra cost of being able to compute efficiently the projected measure onto the subspace, which might require discretization of the space and is therefore not practical in high dimensions. We also proposed a new quadratic cost \(\mathcal{HW}\) that we call Hadamard-Wasserstein, which allows us to define a degenerated cost for which the optimal transport plan converges to a triangular coupling. However, this cost loses many properties compared to \(W_{2}\) or \(GW\), for which we are inclined to use these problems. Indeed, while \(\mathcal{HW}\) is a quadratic cost, it uses a Euclidean norm between the Hadamard product of vectors and requires the two spaces to be the same (in order to have the distance well defined). A work around in the case \(X=\mathbb{R}^{p}\) and \(Y=\mathbb{R}^{q}\) with \(p\leq q\) would be to \"lift\" the vectors in \(\mathbb{R}^{p}\) into vectors in \(\mathbb{R}^{q}\) with padding as it is proposed in [12] or to project the vectors in \(\mathbb{R}^{q}\) on \(\mathbb{R}^{p}\) as in [6]. Yet, for some applications where only the distance/similarity matrices are available, a different strategy still needs to be found. Another concern is the limited invariance properties (only with respect to axial symmetry symmetry in our case). Nevertheless, we expect that such a cost can be of interest in cases where invariance to symmetry is a desired property, such as in [44]. Methodology, C.B., N.C., T.V., F.S., and L.D. ; software, C.B.