State-Space Representations of Deep Neural Networks

Abstract

This letter deals with neural networks as dynamical systems governed by finite difference equations. It shows that the introduction of $k$ -many skip connections into network architectures, such as residual networks and additive dense networks, defines $k$ th order dynamical equations on the layer-wise transformations. Closed-form solutions for the state-space representations of general $k$ th order additive dense networks, where the concatenation operation is replaced by addition, as well as $k$ th order smooth networks, are found. The developed provision endows deep neural networks with an algebraic structure. Furthermore, it is shown that imposing $k$ th order smoothness on network architectures with $d$ -many nodes per layer increases the state-space dimension by a multiple of $k$ , and so the effective embedding dimension of the data manifold by the neural network is $k·d$ -many dimensions. It follows that network architectures of these types reduce the number of parameters needed to maintain the same embedding dimension by a factor of $k^2$ when compared to an equivalent first-order, residual network. Numerical simulations and experiments on CIFAR10, SVHN, and MNIST have been conducted to help understand the developed theory and efficacy of the proposed concepts.