  We propose a differential geometric construction for families of low-rank
covariance matrices, via interpolation on low-rank matrix manifolds. In
contrast with standard parametric covariance classes, these families offer
significant flexibility for problem-specific tailoring via the choice of
"anchor" matrices for the interpolation. Moreover, their low-rank facilitates
computational tractability in high dimensions and with limited data. We employ
these covariance families for both interpolation and identification, where the
latter problem comprises selecting the most representative member of the
covariance family given a data set. In this setting, standard procedures such
as maximum likelihood estimation are nontrivial because the covariance family
is rank-deficient; we resolve this issue by casting the identification problem
as distance minimization. We demonstrate the power of these differential
geometric families for interpolation and identification in a practical
application: wind field covariance approximation for unmanned aerial vehicle
navigation.
