论文标题
集群代数：网络科学和机器学习
Cluster Algebras: Network Science and Machine Learning
论文作者
论文摘要
集群代数最近已成为数学和物理学的重要参与者。在这项工作中，我们通过现代数据科学的角度进行了调查，特别是通过网络科学和机器学习的技术进行了研究。网络分析方法应用于不同突变类型的集群代数的交换图。该分析表明，当图表表示不识别簇之间的排列等效性时，Quiver Exchange图嵌入中出现了优雅的对称性。对于有限的Dynkin类型代数为5等级的种子数量与与此对称性相关的Quivers数量之间的比率是计算的，并以较高的等级猜想。简单的机器学习技术成功地学习了使用种子数据对群集代数进行分类。学习绩效超过相同突变类型的代数和类型之间以及相对于人工生成的数据之间的0.9精度。
Cluster algebras have recently become an important player in mathematics and physics. In this work, we investigate them through the lens of modern data science, specifically with techniques from network science and machine learning. Network analysis methods are applied to the exchange graphs for cluster algebras of varying mutation types. The analysis indicates that when the graphs are represented without identifying by permutation equivalence between clusters an elegant symmetry emerges in the quiver exchange graph embedding. The ratio between number of seeds and number of quivers associated to this symmetry is computed for finite Dynkin type algebras up to rank 5, and conjectured for higher ranks. Simple machine learning techniques successfully learn to classify cluster algebras using the data of seeds. The learning performance exceeds 0.9 accuracies between algebras of the same mutation type and between types, as well as relative to artificially generated data.
