2026/07/22 by Kilian Rueß, Gennadiy Averkov, Florestan Brunck +7
#cs.LG #cs.NE #math.CO
We prove that the maximum of n real numbers is exactly representable by a ReLU network with two hidden layers for every n≤ 10. The constructions are obtained by reducing the problem to exact rational linear algebra: after a symmetry reduction, the necessary cancellations are encoded in finite linear systems over ℚ, which we solve and verify computationally. The representation of max10 has a structured first hidden layer consisting only of pairwise maxima, a feature that allows it to be recursively substituted into larger networks. We use this to show that for every n>10, the maximum maxn can be exactly represented with \lceillog5 (n / 2)\rceil+1 < log5(n) +1.5694 hidden layers. Via the generalized hinging-hyperplane representation [Wang, Sun, IEEE Trans. Inf. Theory 2005], the same depth bound holds for all continuous piecewise-linear functions on ℝd, with d+1 in place of n. In particular, every continuous piecewise-linear function on ℝd for d≤ 9 admits a two-hidden-layer ReLU representation. Our results improve on [Bakaev, Brunck, Hertrich, Stade, Yehudayoff, STOC'26]. In that work, the authors established a two-hidden-layer representation for max5 and an upper bound of \lceillog3 (n-2)\rceil+1 hidden layers for maxn.