In an attempt to better understand structural benefits and generalization power of deep neural networks, we firstly present a novel graph theoretical formulation of neural network models, including fully connected, residual network~(ResNet) and densely connected networks~(DenseNet). Secondly, we extend the error analysis of the population risk for two layer network~\cite{ew2019prioriTwo} and ResNet~\cite{e2019prioriRes} to DenseNet, and show further that for neural networks satisfying certain mild conditions, similar estimates can be obtained. These estimates are a priori in nature since they depend sorely on the information prior to the training process, in particular, the bounds for the estimation errors are independent of the input dimension.

In this project, I've built a neural network architecture with a static execution graph that acts as a dynamic neural network in which connections between various neurons are controlled by the network itself. This is accomplished by manipulating the adjacency matrix representation of the network on a per-neuron basis with cell elements representing a'distance', and masking off connections that are within a threshold. Including a loss term based on the networks sparsity or processing time allows the architecture to optimize its structure for accuracy or speed. Alright, so hopefully I've caught your attention with the title. To begin, I'd like to explain a little behind why I've created this. My educational background is actually in the sciences, just at the junction between chemistry and physics.