  Bayesian learning with Gaussian processes demonstrates encouraging regression
and classification performances in solving computer vision tasks. However,
Bayesian methods on 3D manifold-valued vision data, such as meshes and point
clouds, are seldom studied. One of the primary challenges is how to effectively
and efficiently aggregate geometric features from the irregular inputs. In this
paper, we propose a hierarchical Bayesian learning model to address this
challenge. We initially introduce a kernel with the properties of
geometry-awareness and intra-kernel convolution. This enables geometrically
reasonable inferences on manifolds without using any specific hand-crafted
feature descriptors. Then, we use a Gaussian process regression to organize the
inputs and finally implement a hierarchical Bayesian network for the feature
aggregation. Furthermore, we incorporate the feature learning of neural
networks with the feature aggregation of Bayesian models to investigate the
feasibility of jointly learning on manifolds. Experimental results not only
show that our method outperforms existing Bayesian methods on manifolds but
also demonstrate the prospect of coupling neural networks with Bayesian
networks.
