  Entropy regularization in optimal transport (OT) has been the driver of many
recent interests for Wasserstein metrics and barycenters in machine learning.
It allows to keep the appealing geometrical properties of the unregularized
Wasserstein distance while having a significantly lower complexity thanks to
Sinkhorn's algorithm. However, entropy brings some inherent smoothing bias,
resulting for example in blurred barycenters. This side effect has prompted an
increasing temptation in the community to settle for a slower algorithm such as
log-domain stabilized Sinkhorn which breaks the parallel structure that can be
leveraged on GPUs, or even go back to unregularized OT. Here we show how this
bias is tightly linked to the reference measure that defines the entropy
regularizer and propose debiased Wasserstein barycenters that preserve the best
of both worlds: fast Sinkhorn-like iterations without entropy smoothing.
Theoretically, we prove that the entropic OT barycenter of univariate Gaussians
is a Gaussian and quantify its variance bias. This result is obtained by
extending the differentiability and convexity of entropic OT to sub-Gaussian
measures with unbounded supports. Empirically, we illustrate the reduced
blurring and the computational advantage on various applications.
