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We present a new, unifying approach following some recent developments on thecomplexity of neural networks with piecewise linear activations. We treatneural network layers with piecewise linear activations as tropicalpolynomials, which generalize polynomials in the so-called $(\max, +)$ ortropical algebra, with possibly real-valued exponents. Motivated by thediscussion in (arXiv:1402.1869), this approach enables us to refine their upperbounds on linear regions of layers with ReLU or leaky ReLU activations to$\min\left\{ 2^m, \sum_{j=0}^n \binom{m}{j} \right\}$, where $n, m$ are thenumber of inputs and outputs, respectively. Additionally, we recover theirupper bounds on maxout layers. Our work follows a novel path, exclusively underthe lens of tropical geometry, which is independent of the improvementsreported in (arXiv:1611.01491, arXiv:1711.02114). Finally, we present ageometric approach for effective counting of linear regions using randomsampling in order to avoid the computational overhead of exact countingapproaches

A Tropical Approach to Neural Networks with Piecewise Linear Activations