This paper 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, define k th order dynamical equations on the layer-wise transformations. Closedform 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.
REF explored the algebraic structure of neural networks governed by finite difference equations.
State Space Representations of Deep Neural Networks
Neural Computation, Volume 31, Issue 3, March 2019, p.538-554
Mathematics Computer Science Medicine