Non-parametric threshold for smoothed empirical Wasserstein distance
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Jia-zyjia-SM-EECS-2022.thesis.pdf
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Thesis PDF
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5406b366cfb5711c2591dff2f4ffc152
Author(s)
Jia, Zeyu
Advisor(s)
Polyanskiy, Yury
Rakhlin, Sasha
Date Issued
February 2022
Publisher
Massachusetts Institute of Technology
Abstract
Consider an empirical measure Pš induced by š iid samples from a š-dimensional š¾-subgaussian distribution P. We show that when š¾ < š, the Wasserstein distance šāĀ² (Pā*š©(0, šĀ² š¼ subscript š), P*š© (0, šĀ² š¼ subscript š)) converges at the parametric rate š(1/š), and when š¾ > š, there exists a š¾-subgaussian distribution P such that šāĀ² (Pā *š© (0, šĀ² š¼ subscript š), P* š© (0, šĀ² š¼ subscript š)) = š(1/š). This resolves the open problems in[7], closes the gap between where we get parametric rate and where we do not have parametric rate. Our result provides a complete characterization of the range of parametric rates for subgaussian š.
In addition, when š < š¾, we establish more delicate results about the convergence rate of W2 distance squared. Assuming the distribution is one dimensional, we provide both the lower bound and the upper bound, demonstrating that the rate changes gradually from Ī(1/ā š) to Ī(1/š) as š/š¾ goes from 0 to 1. Moreover, we also establish that š·āā(Pā * š© (0, šĀ² š¼ subscript š)āP * š© (0, šĀ² š¼ subscript š)) = šŖĖ(1/š). These results indicate a dichotomy of the convergence rate between the W2 distance squared and the KL divergence, resulting in the failure of šā-transportation inequality when š < š¾, hence also resolving the open problem in [17] about whether š¾ < š is necessary in proving whether the log-Sobolev inequality holds for P * š© (0, šĀ²).
In addition, when š < š¾, we establish more delicate results about the convergence rate of W2 distance squared. Assuming the distribution is one dimensional, we provide both the lower bound and the upper bound, demonstrating that the rate changes gradually from Ī(1/ā š) to Ī(1/š) as š/š¾ goes from 0 to 1. Moreover, we also establish that š·āā(Pā * š© (0, šĀ² š¼ subscript š)āP * š© (0, šĀ² š¼ subscript š)) = šŖĖ(1/š). These results indicate a dichotomy of the convergence rate between the W2 distance squared and the KL divergence, resulting in the failure of šā-transportation inequality when š < š¾, hence also resolving the open problem in [17] about whether š¾ < š is necessary in proving whether the log-Sobolev inequality holds for P * š© (0, šĀ²).
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Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
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