Shallower ReLU Network Representations via Exact Linear Algebra

We prove that the maximum of n real numbers is exactly representable by a ReLU network with two hidden layers for every n\le 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 \mathbb{Q}, which we solve and verify computationally. The representation of \max_{10} 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 \max_{n} can be exactly represented with \lceil{\log_5 (n / 2)\rceil}+1 < \log_5(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 \mathbb{R}^d, with d+1 in place of n. In particular, every continuous piecewise-linear function on \mathbb{R}^d for d\le 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 \max_{5} and an upper bound of \lceil{\log_3 (n-2)\rceil}+1 hidden layers for \max_{n}.