  Why and how that deep learning works well on different tasks remains a
mystery from a theoretical perspective. In this paper we draw a geometric
picture of the deep learning system by finding its analogies with two existing
geometric structures, the geometry of quantum computations and the geometry of
the diffeomorphic template matching. In this framework, we give the geometric
structures of different deep learning systems including convolutional neural
networks, residual networks, recursive neural networks, recurrent neural
networks and the equilibrium prapagation framework. We can also analysis the
relationship between the geometrical structures and their performance of
different networks in an algorithmic level so that the geometric framework may
guide the design of the structures and algorithms of deep learning systems.
