We investigate the convergence rate of the optimal entropic cost vε to the optimal transport cost as the noise parameter ε ↓ 0. We show that for a large class of cost functions c on Rd×Rd (for which optimal plans are not necessarily unique or induced by a transport map) and compactly supported and L∞ marginals, one has vε − v0 = d/2 εlog(1/ε) + O(ε). Upper bounds are obtained by a block approximation strategy and an integral variant of Alexandrov’s theorem. Under an infinitesimal twist condition on c, i.e. invertibility of ∇^2_xy c(x, y), we get the lower bound by establishing a quadratic detachment of the duality gap in d dimensions thanks to Minty’s trick.

Carlier, G., Pegon, P., Tamanini, L., Convergence rate of general entropic optimal transport costs, <<CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS>>, 2023; 62 (4): N/A/A-N/A. [doi:10.1007/s00526-023-02455-0] [https://hdl.handle.net/10807/232459]

Convergence rate of general entropic optimal transport costs

Tamanini, Luca
2023

Abstract

We investigate the convergence rate of the optimal entropic cost vε to the optimal transport cost as the noise parameter ε ↓ 0. We show that for a large class of cost functions c on Rd×Rd (for which optimal plans are not necessarily unique or induced by a transport map) and compactly supported and L∞ marginals, one has vε − v0 = d/2 εlog(1/ε) + O(ε). Upper bounds are obtained by a block approximation strategy and an integral variant of Alexandrov’s theorem. Under an infinitesimal twist condition on c, i.e. invertibility of ∇^2_xy c(x, y), we get the lower bound by establishing a quadratic detachment of the duality gap in d dimensions thanks to Minty’s trick.
2023
Inglese
Carlier, G., Pegon, P., Tamanini, L., Convergence rate of general entropic optimal transport costs, <<CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS>>, 2023; 62 (4): N/A/A-N/A. [doi:10.1007/s00526-023-02455-0] [https://hdl.handle.net/10807/232459]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10807/232459
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