Machine-Learning Mathematical Structures

He, Yang-Hui

doi:10.48550/arxiv.2101.06317

Cited by 11 publications

(17 citation statements)

References 2 publications

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“…In order to obtain higher dimensional dense lattices with our method, we aim to introduce some machine learning techniques to help us find the elements that we need (e.g., bases of maximal orders in quaternion algebras or the twisting element α). Similar approaches have been taken recently to explore the use of machine learning algorithms to learn properties about algebraic and number theoretic objects [ABH19], [HLO20], [He21].…”

Section: Dimension 16mentioning

confidence: 98%

Even Unimodular Lattices from Quaternion Algebras

Amorós,

Damir,

Hollanti

2021

Preprint

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We review a lattice construction arising from quaternion algebras over number fields and use it to obtain some known extremal and densest lattices in dimensions 8 and 16. The benefit of using quaternion algebras over number fields is that the dimensionality of the construction problem is reduced by 3/4. We explicitly construct the E8 lattice (resp. E 2 8 and Λ16) from infinitely many quaternion algebras over real quadratic (resp. quartic) number fields and we further present a density result on such number fields. By relaxing the extremality condition, we also provide a source for constructing even unimodular lattices in any dimension multiple of 8.

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Section: Dimension 16mentioning

confidence: 98%

Even Unimodular Lattices from Quaternion Algebras

Amorós,

Damir,

Hollanti

2021

Preprint

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“…Even when not reaching 100% accuracy, a rapid and highly accurate NN estimate could reduce practical computations, say, of searching the exact Standard Model within string string, many orders of magnitude faster. Utility aside, the unexpected success of machine learning of algebraic geometry beckons a deeper question: can one machine learn mathematics [24,35]? By this we mean several levels: can ML/AI each column is a defining polynomial.…”

Section: The Landscape Of Mathematicsmentioning

confidence: 99%

From the String Landscape to the Mathematical Landscape: a Machine-Learning Outlook

He¹

2022

Preprint

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“…On the empirical side, a recent publication on machine learning of mathematical structures He [2021] observes that there seem to be a hierarchy of mathematical problems ordered by their amenability to use machine learning (ML) methods, with numerical analysis being the easiest and combinatorics and analytic number theory being the hardest. Note that the latter two deal with properties of discrete structures, even if they apply methods of mathematical analysis.…”

Section: Mathematical Aspect: Are Nn Universal Approximators?mentioning

confidence: 99%

Abstraction, Reasoning and Deep Learning: A Study of the "Look and Say" Sequence

Zadrozny¹

2021

Preprint

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The ability to abstract, count, and use System 2 reasoning are well-known manifestations of intelligence and understanding. In this paper, we argue, using the example of the "Look and Say" puzzle, that although deep neural networks can exhibit high 'competence' (as measured by accuracy) when trained on large data sets (2M examples in our case), they do not show any sign on the deeper understanding of the problem, or what D. Dennett calls 'comprehension'.We report on two sets experiments on the "Look and Say" puzzle data. We view the problem as building a translator from one set of tokens to another. We apply both standard LSTMs and Transformer/Attention-based neural networks, using publicly available machine translation software. We observe that despite the amazing accuracy (on both, training and test data), the performance of the trained programs on the actual L&S sequence is bad.We then discuss a few possible ramifications of this finding and connections to other work, experimental and theoretical. First, from the cognitive science perspective, we argue that we need better mathematical models of abstraction. Second, the classical and more recent results on the universality of neural networks should be re-examined for functions acting on discrete data sets. Mapping on discrete sets usually have no natural continuous extensions. This connects the results on a simple puzzle to more sophisticated results on modeling of mathematical functions, where algebraic functions are more difficult to model than e.g. differential equations. Third, we hypothesize that for problems such as "Look and Say", computing the parity of bitstrings, or learning integer addition, it might be worthwhile to introduce concepts from topology, where continuity is defined without the reference to the concept of distance.

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Machine-Learning Mathematical Structures

Cited by 11 publications

References 2 publications

Even Unimodular Lattices from Quaternion Algebras

Even Unimodular Lattices from Quaternion Algebras

From the String Landscape to the Mathematical Landscape: a Machine-Learning Outlook

Abstraction, Reasoning and Deep Learning: A Study of the "Look and Say" Sequence

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