Convergence of Entropic Schemes for Optimal Transport and Gradient Flows

Abstract

International audienceReplacing positivity constraints by an entropy barrier is popular to approximate solutions of linear programs. In the special case of the optimal transport problem, this technique dates back to the early work of Schr\"odinger. This approach has recently been used successfully to solve optimal transport related problems in several applied fields such as imaging sciences, machine learning and social sciences. The main reason for this success is that, in contrast to linear programming solvers, the resulting algorithms are highly parallelizable and take advantage of the geometry of the computational grid (e.g. an image or a triangulated mesh). The first contribution of this article is the proof of the Γ-convergence of the entropic regularized optimal transport problem towards the Monge-Kantorovich problem for the squared Euclidean norm cost function. This implies in particular the convergence of the optimal entropic regularized transport plan towards an optimal transport plan as the entropy vanishes. Optimal transport distances are also useful to define gradient flows as a limit of implicit Euler steps according to the transportation distance. Our second contribution is a proof that implicit steps according to the entropic regularized distance converge towards the original gradient flow when both the step size and the entropic penalty vanish (in some controlled way)

Fisher information regularization schemes for Wasserstein gradient flows

“…Remark 3 (Comparison with entropic regularization of gradient flow [57]). We also compare our method with the entropic gradient flow studied in [14,57]. A known fact is that when H(ρ) = ρ(x) log ρ(x)dx, the SBP problem has the static formulation [55] SBP(ρ 0 ,…”

“…where α ≥ 0 is a constant and the infimum is over all joint histogram π(x, y) ≥ 0 with marginals ρ 0 (x), ρ 1 (y). In [14,57], the algorithm applies the above static formulation and considers the iterative regularization algorithm for the computation of gradient flow. Our formulation mainly uses the dynamical formulation of SBP, especially its time symmetric version in Proposition 2.…”

“…It is also important to mention that the Fisher information regularization is closely related to the entropic regularization that has been successfully applied in many optimal transport problems [14,32,37,40,56]. There, the Kantorovich formulation based on the joint distribution π(x, y) between two measures is adopted, and the entropic regularization term π(x, y) log π(x, y)dxdy is added to the cost function, so that an iterative projection method, Sinkhorn method or more general Dykstra's algorithm, can be applied with linear convergence.…”

Discretization of flux-limited gradient flows: $\Gamma $-convergence and numerical schemes

“…The core idea of the convergence proof is to study the dual problem for (39), for which the iteration (40) attains a considerably easier form. We refer to [27,9,15] for further discussion of the algorithm, including questions of well-posedness and convergence, in the context of fully discrete approximation of gradient flows.…”

“…The focus has been mainly on image and data science, but the ideas have been applied for numerical approximation of gradient flows as well, see e.g. [27,9]. Here, we develop this approach further to define an efficient scheme for approximation of solutions to flux-limited equations of the type ∂ t ρ + ∇ · ρ a ∇h (ρ) = 0, ρ(0, ·) = ρ 0 .…”