Better Neural Network Expressivity: Subdividing the Simplex

This work studies the expressivity of ReLU neural networks with a focus on their depth. A sequence of previous works showed that $\lceil \log_2(n+1) \rceil$ hidden layers are sufficient to compute all continuous piecewise linear (CPWL) functions on $\mathbb{R}^n$. Hertrich, Basu, Di Summa, and Skutella (NeurIPS'21) conjectured that this result is optimal in the sense that there are CPWL functions on $\mathbb{R}^n$, like the maximum function, that require this depth. We disprove the conjecture and show that $\lceil\log_3(n-1)\rceil+1$ hidden layers are sufficient to compute all CPWL functions on $\mathbb{R}^n$. A key step in the proof is that ReLU neural networks with two hidden layers can exactly represent the maximum function of five inputs. More generally, we show that $\lceil\log_3(n-2)\rceil+1$ hidden layers are sufficient to compute the maximum of $n\geq 4$ numbers. Our constructions almost match the $\lceil\log_3(n)\rceil$ lower bound of Averkov, Hojny, and Merkert (ICLR'25) in the special case of ReLU networks with weights that are decimal fractions. The constructions have a geometric interpretation via polyhedral subdivisions of the simplex into ``easier'' polytopes.

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.