  This paper considers the problem of measure estimation under the barycentric
coding model (BCM), in which an unknown measure is assumed to belong to the set
of Wasserstein-2 barycenters of a finite set of known measures. Estimating a
measure under this model is equivalent to estimating the unknown barycentric
coordinates. We provide novel geometrical, statistical, and computational
insights for measure estimation under the BCM, consisting of three main
results. Our first main result leverages the Riemannian geometry of
Wasserstein-2 space to provide a procedure for recovering the barycentric
coordinates as the solution to a quadratic optimization problem assuming access
to the true reference measures. The essential geometric insight is that the
parameters of this quadratic problem are determined by inner products between
the optimal displacement maps from the given measure to the reference measures
defining the BCM. Our second main result then establishes an algorithm for
solving for the coordinates in the BCM when all the measures are observed
empirically via i.i.d. samples. We prove precise rates of convergence for this
algorithm -- determined by the smoothness of the underlying measures and their
dimensionality -- thereby guaranteeing its statistical consistency. Finally, we
demonstrate the utility of the BCM and associated estimation procedures in
three application areas: (i) covariance estimation for Gaussian measures; (ii)
image processing; and (iii) natural language processing.
