  The Wasserstein space of probability measures is known for its intricate
Riemannian structure, which underpins the Wasserstein geometry and enables
gradient flow algorithms. However, the Wasserstein geometry may not be suitable
for certain tasks or data modalities. Motivated by scenarios where the global
structure of the data needs to be preserved, this work initiates the study of
gradient flows and Riemannian structure in the Gromov-Wasserstein (GW)
geometry, which is particularly suited for such purposes. We focus on the inner
product GW (IGW) distance between distributions on $\mathbb{R}^d$. Given a
functional $\mathsf{F}:\mathcal{P}_2(\mathbb{R}^d)\to\mathbb{R}$ to optimize,
we present an implicit IGW minimizing movement scheme that generates a sequence
of distributions $\{\rho_i\}_{i=0}^n$, which are close in IGW and aligned in
the 2-Wasserstein sense. Taking the time step to zero, we prove that the
discrete solution converges to an IGW generalized minimizing movement (GMM)
$(\rho_t)_t$ that follows the continuity equation with a velocity field $v_t\in
L^2(\rho_t;\mathbb{R}^d)$, specified by a global transformation of the
Wasserstein gradient of $\mathsf{F}$. The transformation is given by a mobility
operator that modifies the Wasserstein gradient to encode not only local
information, but also global structure. Our gradient flow analysis leads us to
identify the Riemannian structure that gives rise to the intrinsic IGW
geometry, using which we establish a Benamou-Brenier-like formula for IGW. We
conclude with a formal derivation, akin to the Otto calculus, of the IGW
gradient as the inverse mobility acting on the Wasserstein gradient. Numerical
experiments validating our theory and demonstrating the global nature of IGW
interpolations are provided.
