arXiv Machine Learning
The Complexity of ReLU Network Synthesis from a Parameter Multiset
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Iterated Narrow Networks: Turing Completeness at Width 4 on Convex Input Spaces and the Complexity of Verification
The paper constructs a width‑4 ReLU network whose iterates simulate any deterministic Turing machine, with polynomial‑time construction and depth per iteration. It shows that reachability for such iterated networks is NP‑complete, NEXP‑complete, or undecidable depending on the time bound, and extends these hardness results to width‑6 networks with common smooth activations and to discretized Neural ODEs in dimension 5.
arXiv Machine Learning
Sep 4
Parameterized Hardness of Zonotope Containment and Neural Network Verification
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 Machine Learning
Jul 17
Random Parameter Noise Does Not Make Exact ReLU Verification Easy
arXiv:2607. 14375v1 Announce Type: cross Abstract: We study exact verification of ReLU networks in an adversarial smoothed model.
arXiv Machine Learning
Oct 9
NP-Hardness of Minimizing Neurons in Two-Hidden-Layer ReLU Neural Networks
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.
arXiv Machine Learning
Oct 21
Polynomial Learning of Positive ReLU Networks under Gaussian Inputs
arXiv:2610. 25116v2 Announce Type: replace Abstract: We study positive, bias-free, one-hidden-layer ReLU networks under standard Gaussian inputs.
arXiv Machine Learning
Aug 26
Every Layer Counts: An Exponential $L_2$ Depth Hierarchy for ReLU Networks
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...