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 $\max_{n}$.

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.

Humanity's Last Exam

Benchmarks are important tools for tracking the rapid advancements in large language model (LLM) capabilities. However, benchmarks are not keeping pace in difficulty: LLMs now achieve over 90\% accuracy on popular benchmarks like MMLU, limiting informed measurement of state-of-the-art LLM capabilities. In response, we introduce Humanity's Last Exam (HLE), a multi-modal benchmark at the frontier of human knowledge, designed to be the final closed-ended academic benchmark of its kind with broad subject coverage. HLE consists of 3,000 questions across dozens of subjects, including mathematics, humanities, and the natural sciences. HLE is developed globally by subject-matter experts and consists of multiple-choice and short-answer questions suitable for automated grading. Each question has a known solution that is unambiguous and easily verifiable, but cannot be quickly answered via internet retrieval. State-of-the-art LLMs demonstrate low accuracy and calibration on HLE, highlighting a significant gap between current LLM capabilities and the expert human frontier on closed-ended academic questions. To inform research and policymaking upon a clear understanding of model capabilities, we publicly release HLE at https://lastexam.ai.