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.

翻译：概率测度的Wasserstein空间以其复杂的黎曼结构而闻名，该结构支撑着Wasserstein几何并使得梯度流算法成为可能。然而，Wasserstein几何可能不适用于某些特定任务或数据模态。受需要保持数据全局结构的场景启发，本研究开创性地在Gromov-Wasserstein（GW）几何中研究梯度流与黎曼结构，该几何特别适用于此类目的。我们专注于$\mathbb{R}^d$上分布之间的内积GW（IGW）距离。给定一个待优化的泛函$\mathsf{F}:\mathcal{P}_2(\mathbb{R}^d)\to\mathbb{R}$，我们提出了一种隐式IGW最小化运动方案，该方案生成一系列分布$\{\rho_i\}_{i=0}^n$，这些分布在IGW意义下相近，并在2-Wasserstein意义下对齐。令时间步长趋于零，我们证明了离散解收敛于一个IGW广义最小化运动（GMM）$(\rho_t)_t$，该运动遵循连续性方程，其速度场$v_t\in L^2(\rho_t;\mathbb{R}^d)$由$\mathsf{F}$的Wasserstein梯度的全局变换所确定。该变换由一个迁移算子给出，该算子修改Wasserstein梯度，使其不仅编码局部信息，也编码全局结构。我们的梯度流分析引导我们识别出产生内在IGW几何的黎曼结构，并利用该结构为IGW建立了一个类Benamou-Brenier公式。最后，我们通过类似于Otto演算的形式推导，得出IGW梯度是作用于Wasserstein梯度的逆迁移算子。本文提供了验证我们理论并展示IGW插值全局性质的数值实验。