  An increasing array of biomedical and computer vision applications requires
the predictive modeling of complex data, for example images and shapes. The
main challenge when predicting such objects lies in the fact that they do not
comply to the assumptions of Euclidean geometry. Rather, they occupy non-linear
spaces, a.k.a. manifolds, where it is difficult to define concepts such as
coordinates, vectors and expected values. In this work, we construct a
non-parametric predictive methodology for manifold-valued objects, based on a
distance modification of the Random Forest algorithm. Our method is versatile
and can be applied both in cases where the response space is a well-defined
manifold, but also when such knowledge is not available. Model fitting and
prediction phases only require the definition of a suitable distance function
for the observed responses. We validate our methodology using simulations and
apply it on a series of illustrative image completion applications, showcasing
superior predictive performance, compared to various established regression
methods.
