  Collecting and aggregating information from several probability measures or
histograms is a fundamental task in machine learning. One of the popular
solution methods for this task is to compute the barycenter of the probability
measures under the Wasserstein metric. However, approximating the Wasserstein
barycenter is numerically challenging because of the curse of dimensionality.
This paper proposes the projection robust Wasserstein barycenter (PRWB) that
has the potential to mitigate the curse of dimensionality. Since PRWB is
numerically very challenging to solve, we further propose a relaxed PRWB
(RPRWB) model, which is more tractable. The RPRWB projects the probability
measures onto a lower-dimensional subspace that maximizes the Wasserstein
barycenter objective. The resulting problem is a max-min problem over the
Stiefel manifold. By combining the iterative Bregman projection algorithm and
Riemannian optimization, we propose two new algorithms for computing the RPRWB.
The complexity of arithmetic operations of the proposed algorithms for
obtaining an $\epsilon$-stationary solution is analyzed. We incorporate the
RPRWB into a discrete distribution clustering algorithm, and the numerical
results on real text datasets confirm that our RPRWB model helps improve the
clustering performance significantly.
