  In this paper we develop a statistical theory and an implementation of deep
learning models. We show that an elegant variable splitting scheme for the
alternating direction method of multipliers optimises a deep learning
objective. We allow for non-smooth non-convex regularisation penalties to
induce sparsity in parameter weights. We provide a link between traditional
shallow layer statistical models such as principal component and sliced inverse
regression and deep layer models. We also define the degrees of freedom of a
deep learning predictor and a predictive MSE criteria to perform model
selection for comparing architecture designs. We focus on deep multiclass
logistic learning although our methods apply more generally. Our results
suggest an interesting and previously under-exploited relationship between deep
learning and proximal splitting techniques. To illustrate our methodology, we
provide a multi-class logit classification analysis of Fisher's Iris data where
we illustrate the convergence of our algorithm. Finally, we conclude with
directions for future research.
