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.
arXiv:2608. 04460v1 Announce Type: cross Abstract: The quantitative analysis of 3D neuronal morphologies requires capturing both graph topology and spatial geometry.
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:2604. 14727v2 Announce Type: replace Abstract: To quantify the geometric capacity of transformers, we develop a tropical-geometric framework for analyzing the spatial partitions induced by conditioned self-attention.
arXiv:2410. 00722v3 Announce Type: replace Abstract: We study convolutional neural networks with monomial activation functions.
arXiv:2504. 06881v2 Announce Type: replace-cross Abstract: Convolutional neural networks (CNNs) are foundational to many state-of-the-art computer vision systems, yet their reliance on multiplication-intensive computations poses challenges for deployment on resource-constrained devices.
The paper examines why standard neural architectures struggle to generalize to longer inputs when solving dynamic programming (DP) problems. It shows that every finite min-plus DP can be represented as a shortest‑path problem on a directed acyclic graph, equivalently as a tropical polynomial whose extended Newton polyhedron captures the decision boundary of the winning path. The authors prove that the graph, polynomial, and polyhedron descriptions form isomorphic semirings at both the formal polynomial and computed function levels, and they demonstrate that the natural dimensionality‑reduction operations in this semiring are neither injective nor closed, revealing structural limitations that hinder length‑generalization.
arXiv:2606. 10806v1 Announce Type: new Abstract: Moonshine is an autonomous agent whose central objective is to generate mathematical conjectures.
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. 11540v1 Announce Type: cross Abstract: We study tropical circuits with scalar multiplication gates, that is, algebraic circuits whose gates implement $\max$, $+$, or multiplication with a positive constant.
arXiv:2608.29530v1 Announce Type: cross Abstract: Modern systems in artificial intelligence (AI) somehow excel in domains for which they seem poorly suited. Intelligence has traditionally been modele...
arXiv:2609.39078v1 Announce Type: new Abstract: Representations are routinely used across machine learning, psychology, and neuroscience to draw inferences about the computations of biological and ar...
Neuro‑Symbolic Geometric Abstraction (NeuSOGA) is a framework that converts raw observations into explicit symbolic mathematical representations. It achieves this by sequentially generating topological and geometric abstractions, using tools such as Euclidean Distance Transforms, Segment Anything, and Implicit Area Splines. The resulting analytical implicit models are interpretable, editable, and support arbitrary‑order smoothness, additive composition, and closed‑form evaluation across diverse sensing modalities.