Deep neural networks (DNNs), particularly those using Rectified Linear Unit (ReLU) activation functions, have achieved remarkable success across diverse machine learning tasks, including image recognition, audio processing, and language modeling. Despite this success, the non-convex nature of DNN loss functions complicates optimization and limits theoretical understanding. In this paper, we highlight how recently developed convex equivalences of ReLU NNs and their connections to sparse signal processing models can address the challenges of training and understanding NNs. Recent research has uncovered several hidden convexities in the loss landscapes of certain NN architectures, notably two-layer ReLU networks and other deeper or varied architectures. This paper seeks to provide an accessible and educational overview that bridges recent advances in the mathematics of deep learning with traditional signal processing, encouraging broader signal processing applications.

In this paper, we study the optimality gap between two-layer ReLU networks regularized with weight decay and their convex relaxations. We show that when the training data is random, the relative optimality gap between the original problem and its relaxation can be bounded by a factor of $O(\sqrt{\log n})$, where $n$ is the number of training samples. A simple application leads to a tractable polynomial-time algorithm that is guaranteed to solve the original non-convex problem up to a logarithmic factor. Moreover, under mild assumptions, we show that with random initialization on the parameters local gradient methods almost surely converge to a point that has low training loss. Our result is an exponential improvement compared to existing results and sheds new light on understanding why local gradient methods work well.