  We consider the problem of graph-matching on a network of 3D shapes with
uncertainty quantification. We assume that the pairwise shape correspondences
are efficiently represented as \emph{functional maps}, that match real-valued
functions defined over pairs of shapes. By modeling functional maps between
nearly isometric shapes as elements of the Lie group $SO(n)$, we employ
\emph{synchronization} to enforce cycle consistency of the collection of
functional maps over the graph, hereby enhancing the accuracy of the individual
maps. We further introduce a tempered Bayesian probabilistic inference
framework on $SO(n)$. Our framework enables: (i) synchronization of functional
maps as maximum-a-posteriori estimation on the Riemannian manifold of
functional maps, (ii) sampling the solution space in our energy based model so
as to quantify uncertainty in the synchronization problem. We dub the latter
\emph{Riemannian Langevin Functional Map (RLFM) Sampler}. Our experiments
demonstrate that constraining the synchronization on the Riemannian manifold
$SO(n)$ improves the estimation of the functional maps, while our RLFM sampler
provides for the first time an uncertainty quantification of the results.
