11 problems

Failure of general universal consistency conjecture. As in the Wasserstein case, universal consistency of $k_n$-NN will not hold in full generality.

Let $\mathsf{pmm}$-spaces be the metric-measure spaces considered in the paper, and let … fails to induce a true metric structure on the space of $\mathsf{pmm}$-spaces. This is mot…

Generalized barycentric projection conjecture. The map $cmathcal{T}_{cgamma^*}$ is a Monge mapping for the $coperatorname{GW}(cmathbb{S},ctilde{cmathbb{X}}_{cgamma^*})$ problem. Th…

Let $g_X$ and $g_Y$ denote the metric functions on the mm-spaces considered in statement (3), and suppose that both are squared Euclidean distances. Squared-Euclidean extension con…

Tightness conjecture. Theorem is tight: there exist $\mu$ and $\nu$ for which optimal correspondence plans for the quadratic Gromov–Wasserstein problem are not maps, but are instea…

Let $x_1<\ldots<x_n$ and $y_1<\ldots<y_n$ be real numbers, let $\alpha>0$, and define … for permutations $\sigma\in S_n$. The assignment problem is to maximize $F_\sigma(x,y)$ over…

A metric-measure coupling model (MCMS) is an object equipped with the structure needed to define both a suitable $k$-WL distance and the Gromov–Wasserstein distance. WL–Gromov–Wass…

Let $p\in[1,\infty)$, and let an $\ell^p$-Gromov-Wasserstein geodesic be a geodesic for the $\ell^p$-Gromov-Wasserstein distance. A geodesic is Hausdorff-bounded if it satisfies th…

For $p\in[1,\infty)$, let $\Gamma^p$ be the collection of all $d_{\mathcal{GW},{p}}^\mathrm{S}$ geodesics, equipped with the uniform metric … Let…

Let $\mu$ and $\nu$ be Gaussian measures on possibly different Euclidean spaces, and consider the Gromov–Wasserstein problem with squared ground distance. The Gaussian-restricted p…

Let $\mu=\mathcal{N}(0,\operatorname{diag}(\alpha))$ and $\nu=\mathcal{N}(0,\operatorname{diag}(\beta))$ be Gaussian measures, and let $GGW_2(\mu,\nu)$ denote the Gromov–Wasserstei…