  In this paper, we investigate the geometric structure of activation spaces of
fully connected layers in neural networks and then show applications of this
study. We propose an efficient approximation algorithm to characterize the
convex hull of massive points in high dimensional space. Based on this new
algorithm, four common geometric properties shared by the activation spaces are
concluded, which gives a rather clear description of the activation spaces. We
then propose an alternative classification method grounding on the geometric
structure description, which works better than neural networks alone.
Surprisingly, this data classification method can be an indicator of
overfitting in neural networks. We believe our work reveals several critical
intrinsic properties of modern neural networks and further gives a new metric
for evaluating them.
