Better Neural Network Expressivity: Subdividing the Simplex

• URL: http://arxiv.org/abs/2505.14338v1
• Date: Tue, 20 May 2025 13:23:57 GMT
• Title: Better Neural Network Expressivity: Subdividing the Simplex
• Authors: Egor Bakaev, Florestan Brunck, Christoph Hertrich, Jack Stade, Amir Yehudayoff,
• Abstract summary: We show that ReLU neural networks with two hidden layers can exactly represent the maximum function of five inputs.<n>Our constructions almost match the $lceillog_3(n)rceil$ lower bound of Averkov, Hojny, and Merkert (ICLR'25) in the special case of ReLU networks with weights that are decimal fractions.
• Score: 4.5030426578394795
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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