  Deep convolutional networks have witnessed unprecedented success in various
machine learning applications. Formal understanding on what makes these
networks so successful is gradually unfolding, but for the most part there are
still significant mysteries to unravel. The inductive bias, which reflects
prior knowledge embedded in the network architecture, is one of them. In this
work, we establish a fundamental connection between the fields of quantum
physics and deep learning. We use this connection for asserting novel
theoretical observations regarding the role that the number of channels in each
layer of the convolutional network fulfills in the overall inductive bias.
Specifically, we show an equivalence between the function realized by a deep
convolutional arithmetic circuit (ConvAC) and a quantum many-body wave
function, which relies on their common underlying tensorial structure. This
facilitates the use of quantum entanglement measures as well-defined
quantifiers of a deep network's expressive ability to model intricate
correlation structures of its inputs. Most importantly, the construction of a
deep ConvAC in terms of a Tensor Network is made available. This description
enables us to carry a graph-theoretic analysis of a convolutional network, with
which we demonstrate a direct control over the inductive bias of the deep
network via its channel numbers, that are related to the min-cut in the
underlying graph. This result is relevant to any practitioner designing a
network for a specific task. We theoretically analyze ConvACs, and empirically
validate our findings on more common ConvNets which involve ReLU activations
and max pooling. Beyond the results described above, the description of a deep
convolutional network in well-defined graph-theoretic tools and the formal
connection to quantum entanglement, are two interdisciplinary bridges that are
brought forth by this work.
