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


Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.