Expressive power of recurrent neural networks
This provides theoretical insight into the efficiency of deep architectures for researchers in neural network theory, but it is incremental as it extends prior work on HT-Networks to TT-Networks.
The paper proves an exponential lower bound on the width of shallow networks equivalent to a class of recurrent neural networks (TT-Networks), showing that processing images patch by patch with an RNN can be exponentially more efficient than a shallow convolutional network.
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.