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Deep convolutional networks have witnessed unprecedented success in variousmachine learning applications. Formal understanding on what makes thesenetworks so successful is gradually unfolding, but for the most part there arestill significant mysteries to unravel. The inductive bias, which reflectsprior knowledge embedded in the network architecture, is one of them. In thiswork, we establish a fundamental connection between the fields of quantumphysics and deep learning. We use this connection for asserting noveltheoretical observations regarding the role that the number of channels in eachlayer of the convolutional network fulfills in the overall inductive bias.Specifically, we show an equivalence between the function realized by a deepconvolutional arithmetic circuit (ConvAC) and a quantum many-body wavefunction, which relies on their common underlying tensorial structure. Thisfacilitates the use of quantum entanglement measures as well-definedquantifiers of a deep network's expressive ability to model intricatecorrelation structures of its inputs. Most importantly, the construction of adeep ConvAC in terms of a Tensor Network is made available. This descriptionenables us to carry a graph-theoretic analysis of a convolutional network, withwhich we demonstrate a direct control over the inductive bias of the deepnetwork via its channel numbers, that are related to the min-cut in theunderlying graph. This result is relevant to any practitioner designing anetwork for a specific task. We theoretically analyze ConvACs, and empiricallyvalidate our findings on more common ConvNets which involve ReLU activationsand max pooling. Beyond the results described above, the description of a deepconvolutional network in well-defined graph-theoretic tools and the formalconnection to quantum entanglement, are two interdisciplinary bridges that arebrought forth by this work.

Deep Learning and Quantum Entanglement: Fundamental Connections with Implications to Network Design