機械学習で図形の構成要素を探る(Machine learning used to probe the building blocks of shapes)

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2023-10-04 インペリアル・カレッジ・ロンドン(ICL)

◆ロンドン帝国大学とノッティンガム大学の数学者は、機械学習を使用して高次元の幾何学の基本的な「原子的な形状」を特定し、数学の進展を急速に加速させる可能性を示しました。
◆彼らはFano varietiesと呼ばれる形状を分析し、量子周期を使用して次元を予測する機械学習モデルを開発しました。このアプローチは新しい数学的洞察を提供し、機械学習モデルの改善にも役立つことが期待されています。

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ファノ多様体の次元を機械学習する Machine learning the dimension of a Fano variety

Tom Coates,Alexander M. Kasprzyk & Sara Veneziale
Nature Communications  Published:08 September 2023
DOI:https://doi.org/10.1038/s41467-023-41157-1

機械学習で図形の構成要素を探る(Machine learning used to probe the building blocks of shapes)

Abstract

Fano varieties are basic building blocks in geometry – they are ‘atomic pieces’ of mathematical shapes. Recent progress in the classification of Fano varieties involves analysing an invariant called the quantum period. This is a sequence of integers which gives a numerical fingerprint for a Fano variety. It is conjectured that a Fano variety is uniquely determined by its quantum period. If this is true, one should be able to recover geometric properties of a Fano variety directly from its quantum period. We apply machine learning to the question: does the quantum period of X know the dimension of X? Note that there is as yet no theoretical understanding of this. We show that a simple feed-forward neural network can determine the dimension of X with 98% accuracy. Building on this, we establish rigorous asymptotics for the quantum periods of a class of Fano varieties. These asymptotics determine the dimension of X from its quantum period. Our results demonstrate that machine learning can pick out structure from complex mathematical data in situations where we lack theoretical understanding. They also give positive evidence for the conjecture that the quantum period of a Fano variety determines that variety.

1504数理・情報
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