arXiv:2607. 21651v1 Announce Type: new Abstract: We prove that the maximum of $n$ real numbers is exactly representable by a ReLU network with two hidden layers for every $n\le 10$.
arXiv:2608. 25221v1 Announce Type: new Abstract: We study exact representations of $\mathrm{MAX}_N(x)=\max{x_1,\ldots,x_N}$ using two-hidden-layer ReLU neural networks.
arXiv:2607. 07778v1 Announce Type: new Abstract: Bubeck, Li and Nagaraj conjectured that, for generic data, any two-layer neural network with $m$ neurons that fits $n$ noisy labels must have Lipschitz constant at least of order $\sqrt{n/m}$, with no restriction on the size of the weights.
arXiv:2608.23877v1 Announce Type: new Abstract: We prove a depth hierarchy for ReLU neural networks in which every additional ReLU layer can save exponentially many neurons. For every $\ell\geq 3$, a...
The paper proves that several decision and approximation problems for ReLU neural networks are computationally hard. For any number of layers λ≥2, deciding whether a network’s output is positive (and thus whether it is surjective) is W[ℓ−1]-hard when parameterized by the input dimension d. In particular, for two-layer networks, the related geometric problem of zonotope non‑containment is W[1]-hard in the ambient dimension, and computing or approximating the Lp‑Lipschitz constant is NP‑hard and W[ℓ−1]-hard with respect to d. The results also show that these problems remain hard when parameterized by the number of layers for constant d, implying that naive enumeration algorithms running in n^{(ℓ−1)d}·poly(N) time are essentially optimal under the Exponential Time Hypothesis.
arXiv:2607. 06781v1 Announce Type: new Abstract: In this work, we investigate the fixed-architecture neural network approximation with explicit parameter bounds and elementary activations.
The paper investigates how to balance approximation accuracy and stability in deep spline superposition networks under a strict layerwise Lipschitz budget. It provides an exact solution to the finite‑depth diagonal balancing problem, shows how to construct spline discretisations that respect the budget, and establishes minimax lower bounds for operators constrained in both first and third derivative norms. The authors also demonstrate that layer errors can accumulate linearly with depth, indicating that the upper bound is not merely a theoretical artifact.
arXiv:2608.24865v1 Announce Type: cross Abstract: Lipschitz constants are a standard way to quantify the sensitivity of neural networks to small input perturbations, but computing them is difficult e...
arXiv:2510. 04060v3 Announce Type: replace-cross Abstract: We establish two related but logically distinct results for shallow ReLU$^k$ neural networks on the unit sphere $\SS^d$.
arXiv:2606. 07728v1 Announce Type: new Abstract: It is well established that ReLU networks define continuous piecewise-linear functions, and that their linear regions are polyhedra in the input space.
arXiv:2607. 10589v1 Announce Type: cross Abstract: In contrast to most studies on neural network approximation theory that characterize results through a single parameter, such as the total number of network parameters, \cite{shen2020deep} pioneered the characterization of approximation rates as a joint function of the width parameter $N$ and the depth parameter $L$, thereby granting greater architectural flexibility.
arXiv:2410. 04907v2 Announce Type: replace-cross Abstract: In this paper we contribute to the frequently studied question of how to decompose a continuous piecewise linear (CPWL) function into a difference of two convex CPWL functions.
arXiv:2608.31157v1 Announce Type: new Abstract: Many parameter-efficient methods generate the parameters of a large neural network from a low-dimensional latent representation. Given an architecture...