Debiased Sinkhorn barycenters
Abstract
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
this debiasing is perfect for Gaussian distributions
with equal variance. Empirically, we illustrate the
reduced blurring and the computational advantage
on various applications.