Abstract
We propose a new statistical model, the spiked transport model, which formalizes the assumption that two probability distributions differ only on a low-dimensional subspace. We study the minimax rate of estimation for the Wasserstein distance under this model and show that this low-dimensional structure can be exploited to avoid the curse of dimensionality. As a byproduct of our minimax analysis, we establish a lower bound showing that, in the absence of such structure, the plug-in estimator is nearly rate-optimal for estimating the Wasserstein distance in high dimension. We also give evidence for a statistical-computational gap and conjecture that any computationally efficient estimator is bound to suffer from the curse of dimensionality.
Funding Statement
JNW gratefully acknowledges its support.
The second author was supported by NSF awards IIS-BIGDATA-1838071, DMS-1712596 and CCF-TRIPODS-1740751; ONR grant N00014-17-1-2147.
Acknowledgements
The authors thank Oded Regev for helpful suggestions.
The first author’s part of this research was conducted while at the Institute for Advanced Study.
Citation
Jonathan Niles-Weed. Philippe Rigollet. "Estimation of Wasserstein distances in the Spiked Transport Model." Bernoulli 28 (4) 2663 - 2688, November 2022. https://doi.org/10.3150/21-BEJ1433