[Paper Review] Rates of Estimation of Optimal Transport Maps using Plug-in Estimators via Barycentric Projections

Nabarun Deb, Promit Ghosal+1 more|arXiv (Cornell University)|Jul 4, 2021

Markov Chains and Monte Carlo MethodsMathematics130 references27 citations

TL;DR

The paper analyzes convergence rates for plug-in estimators of optimal transport maps using barycentric projections and shows smoothing (wavelet/Kernal) can mitigate the curse of dimensionality; it also provides rates for Wasserstein distance estimators and applications to Wasserstein barycenters and independence testing.

ABSTRACT

Optimal transport maps between two probability distributions $μ$ and $ν$ on $\mathbb{R}^d$ have found extensive applications in both machine learning and statistics. In practice, these maps need to be estimated from data sampled according to $μ$ and $ν$. Plug-in estimators are perhaps most popular in estimating transport maps in the field of computational optimal transport. In this paper, we provide a comprehensive analysis of the rates of convergences for general plug-in estimators defined via barycentric projections. Our main contribution is a new stability estimate for barycentric projections which proceeds under minimal smoothness assumptions and can be used to analyze general plug-in estimators. We illustrate the usefulness of this stability estimate by first providing rates of convergence for the natural discrete-discrete and semi-discrete estimators of optimal transport maps. We then use the same stability estimate to show that, under additional smoothness assumptions of Besov type or Sobolev type, wavelet based or kernel smoothed plug-in estimators respectively speed up the rates of convergence and significantly mitigate the curse of dimensionality suffered by the natural discrete-discrete/semi-discrete estimators. As a by-product of our analysis, we also obtain faster rates of convergence for plug-in estimators of $W_2(μ,ν)$, the Wasserstein distance between $μ$ and $ν$, under the aforementioned smoothness assumptions, thereby complementing recent results in Chizat et al. (2020). Finally, we illustrate the applicability of our results in obtaining rates of convergence for Wasserstein barycenters between two probability distributions and obtaining asymptotic detection thresholds for some recent optimal-transport based tests of independence.

Motivation & Objective

• Motivate the estimation of optimal transport maps T0 from samples drawn from μ and ν.
• Develop a unified stability framework for barycentric projections to analyze plug-in estimators.
• Derive rates of convergence for natural discrete-discrete and semi-discrete plug-in estimators.
• Demonstrate how smooth density assumptions (Besov or Sobolev) improve rates and mitigate dimensionality.
• Link results to W2 estimation, Wasserstein barycenters, and independence testing applications.

Proposed method

• Introduce a new stability estimate (Theorem 2.1) for barycentric projections that requires minimal smoothness and does not rely on the existence of an OT map between approximations.
• Derive convergence rates for the natural discrete-discrete and semi-discrete plug-in estimators under Lipschitz OT map (Theorem 2.2).
• Show that Besov Besov-smooth densities yield rates for T0 of the form n^{-(1+s)/(d+2s)} for d ≥ 3 (Theorem 2.4).
• Show that Sobolev-smooth densities with kernel smoothing yield rates of the form m^{-(s+2)/d ∧ 1/2}+n^{-(s+2)/d ∧ 1/2} (Theorem 2.6).
• Establish computable discrete approximations of smoothed measures yielding the same rates (Theorem 2.8).
• Extend the analysis to convergence rates for W2^2(μ,ν) and discuss implications for Wasserstein barycenters and independence testing.

Experimental results

Research questions

• RQ1 What are the rates of convergence for plug-in estimators of the OT map when using barycentric projections from empirical measures?
• RQ2 Can smoothing via Besov or Sobolev density assumptions mitigate the curse of dimensionality in OT map estimation?
• RQ3 What are the convergence rates for plug-in estimators of W2^2(μ,ν) under similar smoothness assumptions?
• RQ4 How do these rates apply to practical OT-related tasks such as Wasserstein barycenters and independence testing?

Key findings

• A new stability bound (Theorem 2.1) for barycentric projections that does not require smoothness or an existing OT map between approximations.
• For discrete-discrete and semi-discrete plug-in estimators with Lipschitz T0 and compactly supported ν, the error rate is m^{-2/d}+n^{-2/d} for d ≥ 4 (Theorem 2.2).
• Under Besov-smooth densities, the rate improves to n^{-(1+s)/(d+2s)} (Theorem 2.4).
• Under Sobolev-smooth densities with kernel smoothing, the rate becomes m^{-(s+2)/d ∧ 1/2}+n^{-(s+2)/d ∧ 1/2} (Theorem 2.6).
• The same smoothing-driven rate improvements hold for estimating W2^2(μ,ν) (Theorem 2.4 and related results).
• Computable discretized smoothed estimators (Theorem 2.8) retain the same rates with an explicit trade-off between statistical and computational complexity.