Finding Kissing Numbers with Game-theoretic Reinforcement Learning
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arXiv
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| Auteurs principaux: | , , , , , , , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866911440792715264 |
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| author | Ma, Chengdong Zhaowei, Théo Tao Li, Pengyu Liu, Minghao Chen, Haojun Mao, Zihao Cheng, Yuan Qi, Yuan Yang, Yaodong |
| author_facet | Ma, Chengdong Zhaowei, Théo Tao Li, Pengyu Liu, Minghao Chen, Haojun Mao, Zihao Cheng, Yuan Qi, Yuan Yang, Yaodong |
| contents | 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 is the local analogue of Hilbert's 18th problem, bridging geometry, number theory, and information theory. Although significant progress has been made through lattices and codes, the irregularities of high-dimensional geometry, dimensional structure variability, and combinatorial explosion beyond Go limit the scalability and generality of existing methods. Here we model the problem as a two-player matrix completion game and train the reinforcement learning system, PackingStar, to play the games. The matrix entries represent pairwise cosines of sphere center vectors. One player fills entries while another corrects suboptimal ones to improve exploration quality, cooperatively maximizing the matrix size, corresponding to the kissing number. These matrices are decomposed into representative substructures, providing diverse bases and structural constraints that steer subsequent games and make extremely large spaces tractable. PackingStar surpasses records from dimensions 25 to 31 and sets new lower bounds for generalized kissing numbers under various angular constraints. It achieves the first breakthrough beyond rational structures from 1971 in 13 dimensions and discovers over 6000 new structures in other dimensions. Notably, some configurations challenge long-held antipodal paradigms, revealing algebraic correspondences with finite simple groups as well as geometric relationships across dimensions. Inspired by these patterns, humans devised further improved constructions. These results demonstrate AI's power to explore high-dimensional spaces beyond human intuition via extreme-scale reinforcement learning and open new pathways for the Kissing Number Problem and broader geometry research. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_13391 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Finding Kissing Numbers with Game-theoretic Reinforcement Learning Ma, Chengdong Zhaowei, Théo Tao Li, Pengyu Liu, Minghao Chen, Haojun Mao, Zihao Cheng, Yuan Qi, Yuan Yang, Yaodong Machine Learning Artificial Intelligence 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 is the local analogue of Hilbert's 18th problem, bridging geometry, number theory, and information theory. Although significant progress has been made through lattices and codes, the irregularities of high-dimensional geometry, dimensional structure variability, and combinatorial explosion beyond Go limit the scalability and generality of existing methods. Here we model the problem as a two-player matrix completion game and train the reinforcement learning system, PackingStar, to play the games. The matrix entries represent pairwise cosines of sphere center vectors. One player fills entries while another corrects suboptimal ones to improve exploration quality, cooperatively maximizing the matrix size, corresponding to the kissing number. These matrices are decomposed into representative substructures, providing diverse bases and structural constraints that steer subsequent games and make extremely large spaces tractable. PackingStar surpasses records from dimensions 25 to 31 and sets new lower bounds for generalized kissing numbers under various angular constraints. It achieves the first breakthrough beyond rational structures from 1971 in 13 dimensions and discovers over 6000 new structures in other dimensions. Notably, some configurations challenge long-held antipodal paradigms, revealing algebraic correspondences with finite simple groups as well as geometric relationships across dimensions. Inspired by these patterns, humans devised further improved constructions. These results demonstrate AI's power to explore high-dimensional spaces beyond human intuition via extreme-scale reinforcement learning and open new pathways for the Kissing Number Problem and broader geometry research. |
| title | Finding Kissing Numbers with Game-theoretic Reinforcement Learning |
| topic | Machine Learning Artificial Intelligence |
| url | https://arxiv.org/abs/2511.13391 |