  We present a novel framework for domain adaptation, whereby both geometric
and statistical differences between a labeled source domain and unlabeled
target domain can be integrated by exploiting the curved Riemannian geometry of
statistical manifolds. Our approach is based on formulating transfer from
source to target as a problem of geometric mean metric learning on manifolds.
Specifically, we exploit the curved Riemannian manifold geometry of symmetric
positive definite (SPD) covariance matrices. We exploit a simple but important
observation that as the space of covariance matrices is both a Riemannian space
as well as a homogeneous space, the shortest path geodesic between two
covariances on the manifold can be computed analytically. Statistics on the SPD
matrix manifold, such as the geometric mean of two matrices can be reduced to
solving the well-known Riccati equation. We show how the Ricatti-based solution
can be constrained to not only reduce the statistical differences between the
source and target domains, such as aligning second order covariances and
minimizing the maximum mean discrepancy, but also the underlying geometry of
the source and target domains using diffusions on the underlying source and
target manifolds. A key strength of our proposed approach is that it enables
integrating multiple sources of variation between source and target in a
unified way, by reducing the combined objective function to a nested set of
Ricatti equations where the solution can be represented by a cascaded series of
geometric mean computations. In addition to showing the theoretical optimality
of our solution, we present detailed experiments using standard transfer
learning testbeds from computer vision comparing our proposed algorithms to
past work in domain adaptation, showing improved results over a large variety
of previous methods.
