A formula for the time derivative of the entropic cost and applications

By Giovanni Conforti, Luca Tamanini. 2019

65 citations recorded by the source corpus.

In the recent years the Schrodinger problem has gained a lot of attention because of the connection, in the small-noise regime, with the Monge-Kantorovich optimal transport problem. Its optimal value, the \emph{entropic cost} $\mathscr{C}_T$, is here deeply investigated. In this paper we study the regularity of $\mathscr{C}_T$ with respect to the parameter $T$ under a curvature condition and explicitly compute its first and second derivative. As applications: - we determine the large-time limit of $\mathscr{C}_T$ and provide sharp exponential convergence rates; we obtain this result not only for the classical Schrodinger problem but also for the recently introduced Mean Field Schrodinger problem [3]; - under quite strong regularity assumptions, we improve the Taylor expansion of $T \mapsto T\mathscr{C}_T$ around $T=0$ from the first to the second order.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/183c8d105163cb7d028fc1bf8997c1b6ac660fc8

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Espen Bernton
2. Marcel Nutz
3. Johannes Wiesel
4. Promit Ghosal
5. Sinho Chewi
6. Aram-Alexandre Pooladian

In Geometry and Topology

1. Marcel Nutz
2. Sinho Chewi
3. Aram-Alexandre Pooladian

Citations in the corpus

• Entropic optimal transport: convergence of potentials (2021, background): “See also [10, 22] for expansions of the entropic transport cost as ε→ 0, which are related to the speed of convergence of (πε).”
• Stability of entropic optimal transport and Schrödinger bridges (2021, background): “Related results can be found in [16, 40] where the expansion of the optimal cost as a function of ε is studied.”