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We establish, for the first time, connections between feedforward neural networks with ReLU activation and tropical geometry --- we show that the family of such neural networks is equivalent to the family of tropical rational maps. Among other things, we deduce that feedforward ReLU neural networks with one hidden layer can be characterized by zonotopes, which serve as building blocks for deeper networks; we relate decision boundaries of such neural networks to tropical hypersurfaces, a major object of study in tropical geometry; and we prove that linear regions of such neural networks correspond to vertices of polytopes associated with tropical rational functions. An insight from our tropical formulation is that a deeper network is exponentially more expressive than a shallow network.
This work tackles the problem of characterizing and understanding the decision boundaries of neural networks with piecewise linear non-linearity activations. We use tropical geometry, a new development in the area of algebraic geometry, to characteri
Deep Convolutional Neural Networks (DCNNs) are currently the method of choice both for generative, as well as for discriminative learning in computer vision and machine learning. The success of DCNNs can be attributed to the careful selection of thei
We study the family of functions that are represented by a linear convolutional neural network (LCN). These functions form a semi-algebraic subset of the set of linear maps from input space to output space. In contrast, the families of functions repr
Deep Convolutional Neural Networks (DCNNs) is currently the method of choice both for generative, as well as for discriminative learning in computer vision and machine learning. The success of DCNNs can be attributed to the careful selection of their
Two networks are equivalent if they produce the same output for any given input. In this paper, we study the possibility of transforming a deep neural network to another network with a different number of units or layers, which can be either equivale