title: Expressive power of recurrent neural networks
abstract: 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.
authors: Valentin Khrulkov Alexander Novikov Ivan Oseledets
hIndex: Ivan Oseledets 39