Estimation of the Gromov–Wasserstein distance of spheres

Question

Let $(X,d_X,\mu_X)$ and $(Y,d_Y,\mu_Y)$ be two metric measure spaces. A probability measure $\mu$ over $X\times Y$ is called a coupling if $(\pi_1)_\sharp \mu=\mu_X$ and $(\pi_2)_\sharp \mu=\mu_Y$. We define the Gromov–Wasserstein distance $d_{\mathcal GW}(X,Y)$ $^{[1]}$ by

$$\frac{1}{2}\inf_\mu(\int \int |d_X(x,x')-d_Y(y,y')|^p\mu(dx\times dy)\mu(dx' \times dy'))^{1/p}$$

where the infimum is taken over all the couplings of $\mu_X$ and $\mu_Y$.

I wonder if there are any computations/estimate for $d_{\mathcal GW}(S^m,S^n)$, where the distances are geodesic distances and measures are uniform measures on spheres.

Reference: [1] Mémoli, F. Found Comput Math (2011) 11: 417. https://doi.org/10.1007/s10208-011-9093-5

Answer 1

I'll preface this by saying that this is not a complete answer.

First, there is a very nice Python package called POT which has calculation of Wasserstein-Gromov distance included (for discrete measures).

I calculated quite a few examples using a quantization method to discretize the uniform distributions on the spheres. Numerical results suggest that the following coupling is optimal (for all $p\geq 1$), even though I cannot prove it:

Let $m > n$ and $\mu_X$ be the uniform distribution on $S_m$ and $\mu_Y$ the uniform distribution on $S_n$. Let $\mu = \mu_X \circ (id, T)^{-1}$, where $T$ is simply a projection of the spherical coordinates. So $(\varphi_1, ..., \varphi_{m-1})$ is mapped to $(\varphi_{m-n+1}, ..., \varphi_{m-1})$.

Notably, it is not even clear to me that the second marginal of $\mu$ again gives the uniform distribution on $S_n$, even though I guess people more familiar with spherical coordinates might see this immediately.

Initially I had hope to prove that this coupling is optimal by showing that any optimizer has to satisfy certain symmetries, which in the end leads to a coupling equivalent to this projection (a unique optimizer is of course impossible).