arXiv:2412. 05109v2 Announce Type: replace Abstract: We derive universal approximation results for the class of (countably) $m$-rectifiable measures.
By Erwin Riegler, Alex B\"uhler, Yang Pan, Helmut B\"olcskei
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
arXiv:2602. 00511v3 Announce Type: replace Abstract: We introduce \emph{Partition of Unity Neural Networks} (PUNNs), a neural-network architecture for multiclass classification based on the classical mathematical notion of a partition of unity.
By Akram Aldroubi
arXiv:2608. 01357v1 Announce Type: new Abstract: Traditional approximation theory measures convergence rates in terms of the number of parameters or degrees of freedom.
By Tong Mao, Jinchao Xu
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
arXiv:2609. 03129v1 Announce Type: cross Abstract: Several classical machine-learning methods, such as KRRs and SVRs, are both computationally and analytically tractable since their estimators either admit closed-form expressions or are obtained by minimizing convex training objectives; neither feature is generally available for deep neural networks.
By Ruiyang Hong, Hrad Ghoukasian, Anastasis Kratsios