A Library of Mirrors: Deep Neural Nets in Low Dimensions are Convex Lasso Models with Reflection Features

We prove that training neural networks on 1-D data is equivalent to solving a convex Lasso problem with a fixed, explicitly defined dictionary matrix of features. The specific dictionary depends on the activation and depth. We consider 2-layer networks with piecewise linear activations, deep narrow ReLU networks with up to 4 layers, and rectangular and tree networks with sign activation and arbitrary depth. Interestingly in ReLU networks, a fourth layer creates features that represent reflections of training data about themselves. The Lasso representation sheds insight to globally optimal networks and the solution landscape.