This paper studies the complexity of functions computed by deep feedforward neural networks with piecewise linear activations, focusing on the number of linear regions they can generate. The authors analyze how deep networks can map different input regions to the same output, enabling them to compute highly complex functions by reusing computations exponentially as the network depth increases. They compare this to shallow networks and show that deep networks can achieve significantly more linear regions, even with a moderate number of hidden layers. The paper provides theoretical results for two types of networks: rectifier and maxout networks. For rectifier networks, they derive a lower bound on the number of linear regions, showing that deep networks can achieve exponential complexity in terms of depth. For maxout networks, they show that the number of linear regions can grow exponentially with the number of layers. The analysis highlights the advantage of depth in neural networks, demonstrating that deep models can capture more complex patterns and generalize better than shallow models. The results contribute to the understanding of the theoretical capabilities of deep learning models and provide insights into the design of more efficient neural architectures.This paper studies the complexity of functions computed by deep feedforward neural networks with piecewise linear activations, focusing on the number of linear regions they can generate. The authors analyze how deep networks can map different input regions to the same output, enabling them to compute highly complex functions by reusing computations exponentially as the network depth increases. They compare this to shallow networks and show that deep networks can achieve significantly more linear regions, even with a moderate number of hidden layers. The paper provides theoretical results for two types of networks: rectifier and maxout networks. For rectifier networks, they derive a lower bound on the number of linear regions, showing that deep networks can achieve exponential complexity in terms of depth. For maxout networks, they show that the number of linear regions can grow exponentially with the number of layers. The analysis highlights the advantage of depth in neural networks, demonstrating that deep models can capture more complex patterns and generalize better than shallow models. The results contribute to the understanding of the theoretical capabilities of deep learning models and provide insights into the design of more efficient neural architectures.