Expressive power of recurrent neural networks

Valentin Khrulkov, Alexander Novikov, Ivan Oseledets

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.

TL;DR: We prove the exponential efficiency of recurrent-type neural networks over shallow networks.

Keywords: Recurrent Neural Networks, Tensor Train, tensor decompositions, expressive power

Data: [CIFAR-10](https://paperswithcode.com/dataset/cifar-10)

Code: [ 2 community implementations](https://paperswithcode.com/paper/?openreview=S1WRibb0Z)

Community Implementations: [ 1 code implementation](https://www.catalyzex.com/paper/arxiv:1711.00811/code)