A note on the Bures-Wasserstein metric

In this brief note, it is shown that the Bures-Wasserstein (BW) metric on the space positive definite matrices lends itself to convex optimization. In other words, the computation of the BW metric can be posed as a convex optimization problem. In turn, this leads to efficient computations of (i) the BW distance between convex subsets of positive definite matrices, (ii) the BW barycenter, and (iii) incorporating BW distance from a given matrix as a convex constraint. Computations are provided for corroboration.