Gradient Flows and Riemannian Structure in the Gromov-Wasserstein Geometry

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$\mathbb {R}^d$, which preserves the angles within the data and serves as a convenient initial setting due to its analytic tractability. Given a functional \textsf{F}:\mathcal {P}_2(\mathbb {R}^d)\rightarrow \mathbb {R}$\textsf{F}:\mathcal {P}_2(\mathbb {R}^d)\rightarrow \mathbb {R}$ to optimize and an initial distribution \rho _0\in \mathcal {P}_2(\mathbb {R}^d)$\rho _0\in \mathcal {P}_2(\mathbb {R}^d)$, we present an implicit IGW minimizing movement scheme that generates a sequence of distributions \{\rho _i\}_{i=0}^n$\{\rho _i\}_{i=0}^n$, which are close in IGW and aligned in the 2-Wasserstein sense. Taking the time step to zero, we prove that the (piecewise constant interpolation of the) discrete solution converges to an IGW generalized minimizing movement (GMM) (\rho _t)_t$(\rho _t)_t$ that follows the continuity equation with a velocity field v_t\in L^2(\rho _t;\mathbb {R}^d)$v_t\in L^2(\rho _t;\mathbb {R}^d)$, specified by a global transformation of the Wasserstein gradient of \textsf{F}$\textsf{F}$ (viz., the gradient of its first variation). The transformation is given by a mobility operator that modifies the Wasserstein gradient to encode not only local information, but also global structure, as expected for the IGW gradient flow. 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 demonstrating the global nature of IGW interpolations are provided to complement the theory.