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.
By Vincent Froese, Moritz Grillo, Christoph Hertrich, Moritz Stargalla
The paper investigates the geometry and closure properties of function spaces realized by ReLU neural networks with fixed architecture. It proves that the minimal number of convex pieces in continuous piecewise-linear functions is lower semicontinuous under pointwise convergence, and uses this to show that shallow scalar ReLU realization spaces are closed in the weighted supremum norm, while providing a counterexample that demonstrates non-closedness for a specific architecture. The work also characterizes the role of a blow‑up residue in non‑closedness and offers new geometric and functional‑analytic tools for analyzing ReLU realization spaces.
By Kathryn A. Lindsey
arXiv:2606. 26705v1 Announce Type: cross Abstract: Feedforward neural network (NN) expressivity is typically studied by emulating optimal basis-expansion schemes.
By Anastasis Kratsios, Simone Brugiapaglia, Bum Jun Kim, Gregory Cousins, Haitz S\'aez de Oc\'ariz Borde
arXiv:2610. 11313v1 Announce Type: new Abstract: A fundamental question in neural network architecture optimization is whether the minimum hidden-neuron count required to approximate a target function within a prescribed tolerance can be computed efficiently.
By Sangrock Lee
arXiv:2607. 14375v1 Announce Type: cross Abstract: We study exact verification of ReLU networks in an adversarial smoothed model.
By Mojtaba Soltanalian
The paper introduces MACCHIATO, a training algorithm that builds a ReLU‑MLP from partial truth‑table data while simultaneously constructing an explicit Boolean circuit over AND, OR, and XOR gates that certifies the network’s computation. The method iteratively projects residuals onto low‑dimensional Boolean classes, compiles the resulting circuit into a ReLU‑MLP, and uses logic minimization and influence‑based variable selection to achieve a six‑layer network with provable truth‑table error bounds. Experiments on synthetic random‑junta tasks show that these certified networks outperform Adam‑trained MLPs in data‑sparse or projection‑aligned regimes and complete faster than flat ESPRESSO in certain settings.
By Hrad Ghoukasian, Anastasis Kratsios