Advancing Geometry with AI: Multi-agent Generation of Polytopes

TL;DR

This work tackles the challenge of constructing polytopes with extremal combinatorial properties, focusing on the Hirsch conjecture and related problems. It introduces the Hopper AI system, a multi-agent search with a neural network-guided move predictor that treats polytope modification as a hopping game. The method generates millions of polytopes and yields new counterexamples to the Hirsch conjecture in dimension $19$ (via a $24$-vertex prismatoid), improves lower bounds for the longest monotone path, and uncovers neighbourly non-cyclic polytopes, demonstrating that AI can aid in exploring highly non-differentiable geometric spaces. The results suggest AI can accelerate mathematical discovery by producing novel, human-distinct constructions in geometry and related optimization contexts.

Abstract

Polytopes are one of the most primitive concepts underlying geometry. Discovery and study of polytopes with complex structures provides a means of advancing scientific knowledge. Construction of polytopes with specific extremal structure is very difficult and time-consuming. Having an automated tool for the generation of such extremal examples is therefore of great value. We present an Artificial Intelligence system capable of generating novel polytopes with very high complexity, whose abilities we demonstrate in three different and challenging scenarios: the Hirsch Conjecture, the k-neighbourly problem and the longest monotone paths problem. For each of these three problems the system was able to generate novel examples, which match or surpass the best previously known bounds. Our main focus was the Hirsch Conjecture, which had remained an open problem for over 50 years. The highly parallel A.I. system presented in this paper was able to generate millions of examples, with many of them surpassing best known previous results and possessing properties not present in the earlier human-constructed examples. For comparison, it took leading human experts over 50 years to handcraft the first example of a polytope exceeding the bound conjectured by Hirsch, and in the decade since humans were able to construct only a scarce few families of such counterexample polytopes. With the adoption of computer-aided methods, the creation of new examples of mathematical objects stops being a domain reserved only for human expertise. Advances in A.I. provide mathematicians with yet another powerful tool in advancing mathematical knowledge. The results presented demonstrate that A.I. is capable of addressing problems in geometry recognized as extremely hard, and also to produce extremal examples different in nature from the ones constructed by humans.

Figures (5)

• Figure 1: An outline of the Hopper algorithm. (a) A polytope $P$ and all the hyperplanes determined by its vertices. (b) Of these regions, adding a point is only possible in the light orange shaded regions. In each of these regions, adding a point yields a combinatorially non-equivalent polytope. (c) After computation of all hyperplanes (red), the neural network determines a probability distribution on these hyperplanes. (d) A region (red) described by a subset of the hyperplanes is selected using the neural network, then a point $c \in R$ is chosen (deterministically). (e) All polytopes obtained by "hopping" a vertex of $P$ to $c$ are returned, degenerate ones are discarded, and non-degenerate ones are evaluated. (f) One of the resulting polytopes. (This 2$d$ picture obscures some subtleties present in higher dimension. Most importantly, individual modifications can have an enormous effect on the combinatorial type of the polytope. Additionally, the selection of the region $R$ in step (c) is more complicated than this picture makes out. Finally, in higher dimensions the number of regions $R$ yielding potential hops is enormous.)
• Figure 2: Multi-agent architecture. The agents (A) read and write polytopes from the shared polytope repository. Simultaneously pairs of polytopes and hyperplanes are written to the shared data repository. The pairs are labeled positively if the hyperplane was part of a boundary of a successful hop region and negatively otherwise.
• Figure 3: (a) The different scales of the constructions given by the Hopper algorithm, versus human examples. We normalize the bottom base facet and record the PCA eigenvalues of the top base facet, then plot the largest eigenvalue versus the smallest eigenvalue at log scale. The green, red, and purple dots correspond to Santos's original construction, the 25 vertex constructions in matschke2015width, and the first member of the family of constructions in matschke2015width (all other members of this family have approximately the same position in this plot). The blue dots are the constructions found by the Hopper algorithm. We can see the examples found by the Hopper algorithm feature several orders of magnitude of difference between the smallest and largest PCA eigenvalues. (b) Evolution of a prismatoid. The axes represent numbers of vertices. The cyan dots represent the states before the prismatoid increases its width to 6, the blue ones correspond to the states after. The orange dot marks the "ascension" to the non-Hirsch realm of width 6 prismatoids. (c) Evolution of another prismatoid. This prismatoid has a low defect (see \ref{['supp::heuristics']}) equal to $11$.
• Figure 4: Multi-objective training. Two fitness functions share the same global minimum corresponding to a desired state. Alternating between the objectives can lead to bypassing a barrier of local minima.
• Figure 5: (a) The architecture of the Hopper brain. The input is a pair -- a polytope and a hyperplane. The outputs are: the main prediction, of whether the input hyperplane was in a boundary of some successful hop region, and the auxiliary prediction -- the defect (see \ref{['supp::heuristics']}) of the input prismatoid. The "arrow block" highlights the fact that the hyperplane data leg does not participate in the prediction of the defect. (b) Training loss of the Hopper brain. As opposed to more traditional scenarios the training is happening in an online fashion. As the population of polytopes in the repository becomes more refined the model can be expected to lose its validity at time. Such change of the regime can be well seen around step $2.77M$.