The paper introduces a computational tropical geometry framework for symbolically analyzing neural networks with tropical activations. It presents an algorithm that computes the network’s linear regions as explicit unions of polyhedra, proves its correctness, and connects the number of linear regions to the monomials in the tropical expression. The authors also define the Hoffman constant to bound distances to the farthest linear region and release the open‑source Julia library TropicalNN.jl to implement these tools, demonstrating their use on proof‑of‑concept examples.
By Paul Lezeau, Thomas Walker, Yueqi Cao, Shiv Bhatia, Anthea Monod
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 $n\le 10$.
By Kilian Rue{\ss}, Gennadiy Averkov, Florestan Brunck, Moritz Grillo, Christoph Hertrich, Georg Loho, Jack Stade, Moritz Stargalla, Matthew Sun, Martin Winter
arXiv:2410. 00722v3 Announce Type: replace Abstract: We study convolutional neural networks with monomial activation functions.
By Vahid Shahverdi, Giovanni Luca Marchetti, Kathl\'en Kohn
arXiv:2406. 02421v2 Announce Type: replace-cross Abstract: Any continuous piecewise-linear function $F\colon \mathbb{R}^{n}\to \mathbb{R}$ can be represented as a linear combination of $\max$ functions of at most $n+1$ affine-linear functions.
By Christoph Koutschan, Anton Ponomarchuk, Josef Schicho
arXiv:2608. 09707v1 Announce Type: cross Abstract: Embedding trained neural networks as surrogates within optimisation problems is an established practice in operations research.
By Yu Liu, Jan Kronqvist, Fabricio Oliveira
arXiv:2602. 06737v2 Announce Type: replace Abstract: We present a generalized framework for the range verification of neural networks featuring non-linear activation functions.
By Noah Schwartz, Chandra Kanth Nagesh, Sriram Sankaranarayanan, Ramneet Kaur, Tuhin Sahai, Susmit Jha
arXiv:2609.25874v1 Announce Type: new
Abstract: Deep neural networks approximate functions by composing affine maps with nonlinear activations, but how composition itself creates approximation power...
By Wentao Huang, Haizhang Zhang
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.
By Zhimao Wang, Amitabh Basu
arXiv:2508. 03867v2 Announce Type: replace-cross Abstract: We introduce a class of algebraic varieties naturally associated with ReLU neural networks, arising from the piecewise linear structure of their outputs across activation regions in input space, and the piecewise multilinear structure in parameter space.
By Yulia Alexandr, Guido Mont\'ufar
arXiv:2606. 09077v1 Announce Type: new Abstract: The Legendre-Fenchel (LF) transform is a fundamental tool in convex analysis and machine learning that maps lower semi-continuous functions to their convex conjugates.
By Basile Plus-Gourdon, Frank Nielsen
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