Gromov-Wasserstein梯度流的概率表示与收敛性
Probabilistic Representation and Convergence of Gromov-Wasserstein Gradient Flows
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中文总结 AI 辅助
本文研究相对熵的内积Gromov–Wasserstein（IGW）梯度流，证明其不满足现有理论的凸性条件，通过局部凸性估计构造梯度流，推导其显式动力学表示与概率形式，并验证其指数收敛于标准高斯测度。
中文摘要 AI 辅助
Wasserstein梯度流与演化偏微分方程、扩散过程密切相关。本文通过研究关于标准高斯测度$γ$的相对熵$\mathsf{H}(\cdot\\|γ)$的内积Gromov–Wasserstein（IGW）梯度流，为建立IGW梯度流的此类关联迈出了第一步。研究首先证明，对任意$\lambda\in \mathbb{R}$，$\mathsf{H}(\cdot\\|γ)$沿广义或修正广义IGW测地线均不具备$\lambda$-凸性，因此不在Zhang等人（2026）提出的现有IGW梯度流理论的适用范围内。本文通过建立合适的局部凸性估计填补了这一空白，该估计可用于构造梯度流并将其拓展至无限时间域。随后，研究得到了所获动力学的逐步显式表示：从偏积分-微分方程出发，推导出一个非线性Fokker–Planck方程，并证明其二阶矩动力学与分布解耦，满足一个自治矩阵常微分方程。这一特性将IGW动力学简化为一个线性、时间非齐次的Fokker–Planck方程，进而得到其概率表示：即一个类似Ornstein–Uhlenbeck过程的线性随机微分方程的时间边缘流。最后，通过证明该梯度流在相对熵意义下指数收敛于$γ$，研究了其渐近行为。
英文摘要
Wasserstein gradient flows are intimately connected with evolution partial differential equations and diffusion processes. We take the first step in developing such connections for inner product Gromov--Wasserstein (IGW) gradient flows by studying the IGW gradient flow of the relative entropy $\mathsf{H}(\cdot\|γ)$ with respect to the standard Gaussian measure $γ$. We first show that $\mathsf{H}(\cdot\|γ)$ fails to be $λ$-convex along generalized or modified generalized IGW geodesics for any $λ\in \mathbb{R}$, and therefore falls outside the scope of the existing IGW gradient flow theory from Zhang et al. (2026). We bridge this gap by establishing a suitable \emph{local} convexity estimate that enables the construction of the gradient flow and its extension to the infinite time horizon. We then obtain increasingly explicit representations of the resulting dynamics. Starting from a partial integro-differential equation, we derive a nonlinear Fokker--Planck equation and show that its second-moment dynamics decouple from the law as they satisfy an autonomous matrix ODE. This reduces the IGW dynamics to a linear, time-inhomogeneous Fokker--Planck equation, yielding a probabilistic representation as the time-marginal flow of a linear stochastic differential equation resembling the Ornstein--Uhlenbeck process. Finally, we study its asymptotic behavior by establishing exponential convergence of the flow to $γ$ in relative entropy.
