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Stability and statistical inference for semidiscrete optimal transport maps

Abstract Number:

2587 

Submission Type:

Contributed Abstract 

Contributed Abstract Type:

Paper 

Participants:

Ritwik Sadhu (1), Ziv Goldfeld (2), Kengo Kato (1)

Institutions:

(1) Cornell University, N/A, (2) Cornell University, Ithaca, NY

Co-Author(s):

Ziv Goldfeld  
Cornell University
Kengo Kato  
Cornell University

First Author:

Ritwik Sadhu  
Cornell University

Presenting Author:

Ritwik Sadhu  
Cornell University

Abstract Text:

We study statistical inference for the optimal transport (OT) map (also known as the Brenier map) from a known absolutely continuous reference distribution onto an unknown finitely discrete target distribution. We derive limit distributions for the $L^p$-error with arbitrary $p \in [1,\infty)$ and for linear functionals of the empirical OT map, together with their moment convergence. The former has a non-Gaussian limit, whose explicit density is derived, while the latter attains asymptotic normality.
For both cases, we also establish consistency of the nonparametric bootstrap. The derivation of our limit theorems relies on new stability estimates of functionals of the OT map with respect to the dual potential vector, which may be of independent interest. We also discuss applications of our limit theorems to the construction of confidence sets for the OT map and inference for a maximum tail correlation.

Keywords:

Bootstrap|functional delta method|Hadamard directional derivative|limit distribution|optimal transport map|semidiscrete optimal transport

Sponsors:

IMS

Tracks:

Statistical Theory

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