Machine Learning meets Number Theory: The Data Science of Birch-Swinnerton-Dyer
Empirical analysis is often the first step towards the birth of a conjecture. This is the case of the Birch-Swinnerton-Dyer (BSD) Conjecture describing the rational points on an elliptic curve, one of the most celebrated unsolved problems in mathematics. Here we extend the original empirical approach, to the analysis of the Cremona database of quantities relevant to BSD, inspecting more than 2.5 million elliptic curves by means of the latest techniques in data science, machine-learning and topological data analysis. Key quantities such as rank, Weierstrass coefficients, period, conductor, Tamagawa number, regulator and order of the Tate-Shafarevich group give rise to a high-dimensional point-cloud whose statistical properties we investigate. We reveal patterns and distributions in the rank versus Weierstrass coefficients, as well as the Beta distribution of the BSD ratio of the quantities. Via gradient boosted trees, machine learning is applied in finding inter-correlation amongst the various quantities. We anticipate that our approach will spark further research on the statistical properties of large datasets in Number Theory and more in general in pure Mathematics.
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Laura Alessandretti (edit)
Andrea Baronchelli (edit)
Yang-Hui He (add twitter)
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arxiv_cs_LG: Machine Learning meets Number Theory: The Data Science of Birch-Swinnerton-Dyer. Laura Alessandretti, Andrea Baronchelli, and Yang-Hui He https://t.co/wk9FGI96tZ
colliand: A recent post on arXiv with a mashup of #datascience and #MachineLearning ideas to explore and generate insights into #numbertheory: https://t.co/2RaE2mQ5FO
arxivml: "Machine Learning meets Number Theory: The Data Science of Birch-Swinnerton-Dyer", Laura Alessandretti, Andrea Baro… https://t.co/2JMaaitgVW
a_baronca: Latest out: "Machine Learning meets Number Theory: The Data Science of Birch-Swinnerton-Dyer" Pure mathematicians have put together huge, extremely relevant, datasets. Time to dust them off and make maths experimental again! https://t.co/8fqaDQBV1g w @lau_retti and Yang-Hui He https://t.co/lr7aEx8MsG
HigherGeometer: I'm not 100% convinced, but I laughed at this. Laura Alessandretti, Andrea Baronchelli, Yang-Hui He "Machine Learning meets Number Theory: The Data Science of Birch-Swinnerton-Dyer" https://t.co/ouJr9uxY08 https://t.co/KGjBGqmqsu
muktabh: RT @StatsPapers: Machine Learning meets Number Theory: The Data Science of Birch-Swinnerton-Dyer. https://t.co/98owxj7aHW
kcgcse: Machine Learning meets Number Theory: The Data Science of Birch-Swinnerton-Dyer https://t.co/cDS323u8Qf @KCG #kcgcollege #AnnaUniversity #College #ComputerScience @KCGtechnology.
mathNTb: Laura Alessandretti, Andrea Baronchelli, Yang-Hui He : Machine Learning meets Number Theory: The Data Science of Birch-Swinnerton-Dyer https://t.co/Zkd0TvsSBP https://t.co/NvgxRVEhT0
StatsPapers: Machine Learning meets Number Theory: The Data Science of Birch-Swinnerton-Dyer. https://t.co/98owxj7aHW
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