We study regularized deep neural networks and introduce an analytic framework to characterize the structure of the hidden layers. We show that a set of optimal hidden layer weight matrices for a norm regularized deep neural network training problem can be explicitly found as the extreme points of a convex set. For two-layer linear networks, we first formulate a convex dual program and prove that strong duality holds. We then extend our derivations to prove that strong duality also holds for certain deep networks. In particular, for linear deep networks, we show that each optimal layer weight matrix is rank-one and aligns with the previous layers when the network output is scalar. We also extend our analysis to the vector outputs and other convex loss functions. More importantly, we show that the same characterization can also be applied to deep ReLU networks with rank-one inputs, where we prove that strong duality still holds and optimal layer weight matrices are rank-one for scalar output networks. As a corollary, we prove that norm regularized deep ReLU networks yield spline interpolation for one-dimensional datasets which was previously known only for two-layer networks. We then verify our theoretical results via several numerical experiments.

本文研究正则化深度神经网络及其隐层结构，通过凸分析框架构建问题的最优隐层权重，证明For深度ReLU网络，权重矩阵与之前的层通过对偶对齐，并给出了数据为基态或白话时的权重的解析解。同时，该研究也可以甚至适用于具有批归一化架构的深度神经网络，并给出了“神经坍塌”现象的完整解释。

通过凸对偶揭示深度神经网络的结构