The paper introduces a new neural network architecture that represents mixed‑monotone functions as the difference of two monotone subnetworks, enabling tight output enclosures via only two forward passes of opposite corners of an input interval. The enclosure size is a closed‑form function of the network weights, allowing direct minimization during training. Experiments on regression and dynamical‑system benchmarks show that the trained networks achieve enclosure sizes within 0.5% of a proven lower bound while maintaining prediction accuracy, and the method is significantly faster and often tighter than existing techniques such as CROWN, α‑CROWN, and zonotope propagation.
By Jonas Fingerhut, Lukas Koller, Matthias Althoff
arXiv:2607. 21761v1 Announce Type: cross Abstract: We prove function-theoretic analogues of a quantitative result of Hodges on extracting the order property from a sufficiently large 2-tree coded in a binary relation.
By G Conant, C Terry
The paper introduces the concept of supermodular rank, defined as the minimal number of supermodular components required to express a function on a lattice. It characterizes the maximum possible rank and the structure of functions with a fixed rank, and analogously defines submodular rank. Leveraging submodular decompositions, the authors propose an algorithm that partitions a set‑function optimization problem into submodular subproblems, thereby improving approximation guarantees for monotone maximization and ratio minimization while incurring a computational cost tied to the submodular rank.
By Rishi Sonthalia, Anna Seigal, Guido Montufar
The paper introduces recurrent Graph Neural Networks (GNNs) that use set-based aggregation and establishes conditions that can be verified directly from the network weights. It proves a two‑directional equivalence between these networks and the Boolean closure of reachability and safety properties, corresponding to the fragment BΣ◦₁ of the modal μ‑calculus. This equivalence allows for verifiable symbolic explanations of networks that satisfy the identified conditions, without relying on counting logic or external halting signals.
By Blai Bonet
arXiv:2607. 15107v1 Announce Type: new Abstract: This paper develops a categorical framework -- Learning in Infinitesimal Non-Compositional Sketches (LINCS) -- as the repair of non-compositionality: failures of diagrams to factor through quotient sketches lifted to the tangent category setting.
By Sridhar Mahadevan
arXiv:2510. 15814v2 Announce Type: replace-cross Abstract: Universality results for equivariant neural networks remain rare.
By Marco Pacini, Mircea Petrache, Bruno Lepri, Shubhendu Trivedi, Robin Walters