Finding Kissing Numbers with Game-theoretic Reinforcement Learning

2025-11-17

AI Summary

This work addresses the kissing number problem—the maximum number of non-overlapping unit spheres that can simultaneously touch a central unit sphere in high-dimensional Euclidean space. Conventional approaches based on lattice theory and coding theory suffer from combinatorial explosion beyond eight dimensions, severely limiting scalability. To overcome this, we reformulate kissing configuration search as a two-player matrix-filling game, where states are represented by cosine similarities among sphere-center vectors. We propose PackingStar, a cooperative deep reinforcement learning framework for geometric exploration. This is the first method to integrate game-theoretic modeling with deep RL for discrete geometric optimization. Our approach achieves the first improvement in the 13-dimensional kissing number since 1971, surpasses all previously known human-constructed records in dimensions 25–31, and discovers over 6,000 new optimal or near-optimal configurations—including the 25-dimensional Leech lattice—demonstrating the efficacy and frontier potential of AI-driven discovery in high-dimensional discrete geometry.

Technology Category

• Search and Optimization: Learning to Search
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Machine Learning: Learning with Manifolds

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Economics, Online Markets and Human Computation: Uses of LLMs and GenAI for marketplace design, bidding, and strategic interactions
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Since Isaac Newton first studied the Kissing Number Problem in 1694, determining the maximal number of non-overlapping spheres around a central sphere has remained a fundamental challenge. This problem represents the local analogue of Hilbert's 18th problem on sphere packing, bridging geometry, number theory, and information theory. Although significant progress has been made through lattices and codes, the irregularities of high-dimensional geometry and exponentially growing combinatorial complexity beyond 8 dimensions, which exceeds the complexity of Go game, limit the scalability of existing methods. Here we model this problem as a two-player matrix completion game and train the game-theoretic reinforcement learning system, PackingStar, to efficiently explore high-dimensional spaces. The matrix entries represent pairwise cosines of sphere center vectors; one player fills entries while another corrects suboptimal ones, jointly maximizing the matrix size, corresponding to the kissing number. This cooperative dynamics substantially improves sample quality, making the extremely large spaces tractable. PackingStar reproduces previous configurations and surpasses all human-known records from dimensions 25 to 31, with the configuration in 25 dimensions geometrically corresponding to the Leech lattice and suggesting possible optimality. It achieves the first breakthrough beyond rational structures from 1971 in 13 dimensions and discovers over 6000 new structures in 14 and other dimensions. These results demonstrate AI's power to explore high-dimensional spaces beyond human intuition and open new pathways for the Kissing Number Problem and broader geometry problems.

Problem

• Determining the maximum non-overlapping spheres around a central sphere
• Overcoming high-dimensional geometric irregularities and combinatorial complexity
• Exploring kissing number configurations beyond human-known records

Innovation

• Game-theoretic reinforcement learning for high-dimensional spaces
• Two-player matrix completion game modeling sphere arrangements
• Cooperative dynamics maximizing matrix size for kissing numbers