TY - JOUR
T1 - Gradient estimates for the Schrödinger potentials: convergence to the Brenier map and quantitative stability
AU - Chiarini, Alberto
AU - Conforti, Giovanni
AU - Greco, Giacomo
AU - Tamanini, Luca
PY - 2023
Y1 - 2023
N2 - We show convergence of the gradients of the Schrödinger potentials to the Brenier map in the small-time limit under general assumptions on the marginals, which allow for unbounded densities and supports. Furthermore, we provide novel quantitative stability estimates for the optimal values and optimal couplings for the Schrödinger problem (SP), that we express in terms of a negative order weighted homogeneous Sobolev norm. The latter encodes the linearized behavior of the 2-Wasserstein distance between the marginals. The proofs of both results highlight for the first time the relevance of gradient bounds for Schrödinger potentials, that we establish here in full generality, in the analysis of the short-time behavior of Schrödinger bridges. Finally, we discuss how our results translate into the framework of quadratic Entropic Optimal Transport, that is a version of SP more suitable for applications in machine learning and data science.
AB - We show convergence of the gradients of the Schrödinger potentials to the Brenier map in the small-time limit under general assumptions on the marginals, which allow for unbounded densities and supports. Furthermore, we provide novel quantitative stability estimates for the optimal values and optimal couplings for the Schrödinger problem (SP), that we express in terms of a negative order weighted homogeneous Sobolev norm. The latter encodes the linearized behavior of the 2-Wasserstein distance between the marginals. The proofs of both results highlight for the first time the relevance of gradient bounds for Schrödinger potentials, that we establish here in full generality, in the analysis of the short-time behavior of Schrödinger bridges. Finally, we discuss how our results translate into the framework of quadratic Entropic Optimal Transport, that is a version of SP more suitable for applications in machine learning and data science.
KW - Curvature lower bounds
KW - entropic regularization
KW - Schrodinger potentials
KW - optimal transport
KW - quantitative stability
KW - gradient estimates
KW - Curvature lower bounds
KW - entropic regularization
KW - Schrodinger potentials
KW - optimal transport
KW - quantitative stability
KW - gradient estimates
UR - http://hdl.handle.net/10807/259301
U2 - 10.1080/03605302.2023.2215527
DO - 10.1080/03605302.2023.2215527
M3 - Article
SN - 0360-5302
VL - 48
SP - 895
EP - 943
JO - Communications in Partial Differential Equations
JF - Communications in Partial Differential Equations
ER -