Asymptotics of smoothed Wasserstein distances in the small noise regime

Yunzi Ding; Jonathan Niles-Weed

Asymptotics of smoothed Wasserstein distances in the small noise regime

Yunzi Ding, Jonathan Niles-Weed

Mathematics

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

Abstract

We study the behavior of the Wasserstein-2 distance between discrete measures µ and ν in R^d when both measures are smoothed by small amounts of Gaussian noise. This procedure, known as Gaussian-smoothed optimal transport, has recently attracted attention as a statistically attractive alternative to the unregularized Wasserstein distance. We give precise bounds on the approximation properties of this proposal in the small noise regime, and establish the existence of a phase transition: we show that, if the optimal transport plan from µ to ν is unique and a perfect matching, there exists a critical threshold such that the difference between W₂(µ, ν) and the Gaussian-smoothed OT distance W₂(µ ∗ N_σ, ν ∗ N_σ) scales like exp(-c/σ²) for σ below the threshold, and scales like σ above it. These results establish that for σ sufficiently small, the smoothed Wasserstein distance approximates the unregularized distance exponentially well.

Original language	English (US)
Title of host publication	Advances in Neural Information Processing Systems 35 - 36th Conference on Neural Information Processing Systems, NeurIPS 2022
Editors	S. Koyejo, S. Mohamed, A. Agarwal, D. Belgrave, K. Cho, A. Oh
Publisher	Neural information processing systems foundation
ISBN (Electronic)	9781713871088
State	Published - 2022
Event	36th Conference on Neural Information Processing Systems, NeurIPS 2022 - New Orleans, United States Duration: Nov 28 2022 → Dec 9 2022

Publication series

Name	Advances in Neural Information Processing Systems
Volume	35
ISSN (Print)	1049-5258

Conference

Conference	36th Conference on Neural Information Processing Systems, NeurIPS 2022
Country/Territory	United States
City	New Orleans
Period	11/28/22 → 12/9/22

ASJC Scopus subject areas

Computer Networks and Communications
Information Systems
Signal Processing

Cite this

Ding, Y., & Niles-Weed, J. (2022). Asymptotics of smoothed Wasserstein distances in the small noise regime. In S. Koyejo, S. Mohamed, A. Agarwal, D. Belgrave, K. Cho, & A. Oh (Eds.), Advances in Neural Information Processing Systems 35 - 36th Conference on Neural Information Processing Systems, NeurIPS 2022 (Advances in Neural Information Processing Systems; Vol. 35). Neural information processing systems foundation.

Asymptotics of smoothed Wasserstein distances in the small noise regime. / Ding, Yunzi; Niles-Weed, Jonathan.
Advances in Neural Information Processing Systems 35 - 36th Conference on Neural Information Processing Systems, NeurIPS 2022. ed. / S. Koyejo; S. Mohamed; A. Agarwal; D. Belgrave; K. Cho; A. Oh. Neural information processing systems foundation, 2022. (Advances in Neural Information Processing Systems; Vol. 35).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

Ding, Y & Niles-Weed, J 2022, Asymptotics of smoothed Wasserstein distances in the small noise regime. in S Koyejo, S Mohamed, A Agarwal, D Belgrave, K Cho & A Oh (eds), Advances in Neural Information Processing Systems 35 - 36th Conference on Neural Information Processing Systems, NeurIPS 2022. Advances in Neural Information Processing Systems, vol. 35, Neural information processing systems foundation, 36th Conference on Neural Information Processing Systems, NeurIPS 2022, New Orleans, United States, 11/28/22.

Ding Y, Niles-Weed J. Asymptotics of smoothed Wasserstein distances in the small noise regime. In Koyejo S, Mohamed S, Agarwal A, Belgrave D, Cho K, Oh A, editors, Advances in Neural Information Processing Systems 35 - 36th Conference on Neural Information Processing Systems, NeurIPS 2022. Neural information processing systems foundation. 2022. (Advances in Neural Information Processing Systems).

Ding, Yunzi ; Niles-Weed, Jonathan. / Asymptotics of smoothed Wasserstein distances in the small noise regime. Advances in Neural Information Processing Systems 35 - 36th Conference on Neural Information Processing Systems, NeurIPS 2022. editor / S. Koyejo ; S. Mohamed ; A. Agarwal ; D. Belgrave ; K. Cho ; A. Oh. Neural information processing systems foundation, 2022. (Advances in Neural Information Processing Systems).

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Asymptotics of smoothed Wasserstein distances in the small noise regime

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