Faster Wasserstein Distance Estimation with the Sinkhorn Divergence
This work addresses computational efficiency for researchers and practitioners in machine learning and statistics who need fast non-parametric distribution comparisons, but it is incremental as it builds on existing entropic regularization methods.
The paper tackles the problem of estimating the squared Wasserstein distance between probability distributions by proposing the Sinkhorn divergence estimator, which allows higher regularization levels of order ε^{1/2} and leads to improved computational complexity bounds and a strong speedup in practice.
The squared Wasserstein distance is a natural quantity to compare probability distributions in a non-parametric setting. This quantity is usually estimated with the plug-in estimator, defined via a discrete optimal transport problem which can be solved to $ε$-accuracy by adding an entropic regularization of order $ε$ and using for instance Sinkhorn's algorithm. In this work, we propose instead to estimate it with the Sinkhorn divergence, which is also built on entropic regularization but includes debiasing terms. We show that, for smooth densities, this estimator has a comparable sample complexity but allows higher regularization levels, of order $ε^{1/2}$, which leads to improved computational complexity bounds and a strong speedup in practice. Our theoretical analysis covers the case of both randomly sampled densities and deterministic discretizations on uniform grids. We also propose and analyze an estimator based on Richardson extrapolation of the Sinkhorn divergence which enjoys improved statistical and computational efficiency guarantees, under a condition on the regularity of the approximation error, which is in particular satisfied for Gaussian densities. We finally demonstrate the efficiency of the proposed estimators with numerical experiments.