  We propose a robust and scalable procedure for general optimization and
inference problems on manifolds leveraging the classical idea of
`median-of-means' estimation. This is motivated by ubiquitous examples and
applications in modern data science in which a statistical learning problem can
be cast as an optimization problem over manifolds. Being able to incorporate
the underlying geometry for inference while addressing the need for robustness
and scalability presents great challenges. We address these challenges by first
proving a key lemma that characterizes some crucial properties of geometric
medians on manifolds. In turn, this allows us to prove robustness and tighter
concentration of our proposed final estimator in a subsequent theorem. This
estimator aggregates a collection of subset estimators by taking their
geometric median over the manifold. We illustrate bounds on this estimator via
calculations in explicit examples. The robustness and scalability of the
procedure is illustrated in numerical examples on both simulated and real data
sets.
