On the Depth of Monotone ReLU Neural Networks and ICNNs

We study two models of ReLU neural networks: monotone networks (ReLU$^+$) and input convex neural networks (ICNN). Our focus is on expressivity, mostly in terms of depth, and we prove the following lower bounds. For the maximum function MAX$_n$ computing the maximum of $n$ real numbers, we show that ReLU$^+$ networks cannot compute MAX$_n$, or even approximate it. We prove a sharp $n$ lower bound on the ICNN depth complexity of MAX$_n$. We also prove depth separations between ReLU networks and ICNNs; for every $k$, there is a depth-2 ReLU network of size $O(k^2)$ that cannot be simulated by a depth-$k$ ICNN. The proofs are based on deep connections between neural networks and polyhedral geometry, and also use isoperimetric properties of triangulations.