Gradient Flows of Potential Energies in the Geometry of Sinkhorn Divergences

Abstract

We analyze the gradient flow of a potential energy in the space of probability measures when we substitute the optimal transport geometry with a geometry based on Sinkhorn divergences, a debiased version of entropic optimal transport. This gradient flow appears formally as the limit of the minimizing movement scheme, a.k.a. JKO scheme, when the squared Wasserstein distance is substituted by the Sinkhorn divergence. We prove well-posedness and stability of the flow, and that, in the long term, the energy always converges to its minimal value. The analysis is based on a change of variable to study the flow in a Reproducing Kernel Hilbert Space, in which the evolution is no longer a gradient flow but described by a monotone operator. Under a restrictive assumption we prove the convergence of our modified JKO scheme towards this flow as the time step vanishes. We also provide numerical illustrations of the intriguing properties of this newly defined gradient flow.

Figures (7)

• Figure 1: Illustration of the change of variables.
• Figure 2: Numerical simulation (see \ref{['sec:numerics']} for details of the implementation) of the Sinkhorn potential flow on a 3-point space embedded on the RKHS sphere, illustrating the structure of constrained rotation. The white to blue heat map corresponds to the potential energy $\Tilde E$.
• Figure 3: Summary of limit results obtained the present work.
• Figure 4: Sinkhorn potential flows (red for the Eulerian discretization at the top, orange for the Lagrangian one in the middle) for $V(x) = x^2$ (blue plot) on $\mathcal{X} = [0, 1]$ compared to the Wasserstein flow (black, at the bottom), all with the same Gaussian initial condition. For the Sinkhorn flows, the ticks on the x-axis are spaced by $\sqrt{\varepsilon}$ to give a sense of the characteristic distance. In the Lagrangian discretization, we represent the empirical histogram of the particles. The Wasserstein flow is computed in closed form, then discretized to match the other figures.
• Figure 5: Lagrangian flows in 2D space for $V = \left\lVert\cdot\right\rVert^2$ (white to blue heat map), for the Sinkhorn divergence with $\varepsilon =0.04$ (top) vs. the Wasserstein distance (bottom). For the latter, the grid lines are spaced by $\sqrt{\varepsilon}$ to give a sense of the characteristic distance.

Theorems & Definitions (73)

• Theorem 1.1: feydy19
• Lemma 1
• proof
• Remark 1: The flow looks unstable
• Remark 2: Other energies $E$
• Remark 3: Weak formulation of a constrained gradient flow on a Riemannian manifold
• Remark 4: Link with the metric theory
• Proposition 1: RGSD
• Proposition 2: RGSD
• Remark 5