Deep neural networks are surprisingly efficient at solving practical tasks,
 but the theory behind this phenomenon is only starting to catch up with
 the practice. Numerous works show that depth is the key to this efficiency.
 A certain class of deep convolutional networks – namely those that correspond
 to the Hierarchical Tucker (HT) tensor decomposition – has been
 proven to have exponentially higher expressive power than shallow networks.
 I.e. a shallow network of exponential width is required to realize
 the same score function as computed by the deep architecture. In this paper,
 we prove the expressive power theorem (an exponential lower bound on
 the width of the equivalent shallow network) for a class of recurrent neural
 networks – ones that correspond to the Tensor Train (TT) decomposition.
 This means that even processing an image patch by patch with an RNN
 can be exponentially more efficient than a (shallow) convolutional network
 with one hidden layer. Using theoretical results on the relation between
 the tensor decompositions we compare expressive powers of the HT- and
 TT-Networks. We also implement the recurrent TT-Networks and provide
 numerical evidence of their expressivity.
We prove the exponential efficiency of recurrent-type neural networks over shallow networks. The authors compare the complexity of tensor train networks with networks structured by CP decomposition