Machine Learning for Hilbert Series

Hirst, Edward

doi:10.48550/arxiv.2203.06073

Cited by 3 publications

(3 citation statements)

References 58 publications

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“…The work in [30,57] initiates the application of machine learning techniques to these objects, with the aim that from the first terms of the series' Taylor expansions one can extract information about the underlying variety from the HS closed forms from (4.2). With this information one can then compute all higher terms, and hence information about any order BPS operators from just information on the first few, as well information about the physical theory's moduli space.…”

Section: Hilbert Seriesmentioning

confidence: 99%

Machine Learning Algebraic Geometry for Physics

Bao¹,

He²,

Elli³

et al. 2022

Preprint

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We review some recent applications of machine learning to algebraic geometry and physics. Since problems in algebraic geometry can typically be reformulated as mappings between tensors, this makes them particularly amenable to supervised learning. Additionally, unsupervised methods can provide insight into the structure of such geometrical data. At the heart of this programme is the question of how geometry can be machine learned, and indeed how AI helps one to do mathematics. This is a chapter contribution to the book Machine learning and Algebraic Geometry, edited by A. Kasprzyk et al.

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Section: Hilbert Seriesmentioning

confidence: 99%

Machine Learning Algebraic Geometry for Physics

Bao¹,

He²,

Elli³

et al. 2022

Preprint

Self Cite

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“…The techniques for ML range from nonlinear function fitting and landscape searching, to clustering and pattern recognition; and in [24][25][26][27][28] they were first introduced to the string landscape. Further to their success in the algebraic geometry sector of string theory [29][30][31][32][33][34][35][36][37][38][39][40] and the related highenergy physics [41][42][43][44][45][46], strong results from ML application have also been seen in various fields of mathematics [38,[47][48][49][50][51][52][53][54][55]. Particularly relevant to the present context are [56] where the study of ML on algebraic structures was initiated, [57] where ML was applied to quiver mutation, as well as [58] where ML was utilized in classification problems in commutative algebra and [59] where learning strategies were imposed in the key step of Buchberger algorithm in algebraic geometry.…”

Section: Introductionmentioning

confidence: 99%

Cluster Algebras: Network Science and Machine Learning

Dechant¹,

He²,

Heyes³

et al. 2022

Preprint

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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 differentiate cluster algebras from their 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.

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“…In Ref. [18][19][20][21][22], the authors utilized deep layers satisfying a specific recurrence relation. In this case, experimental or observational data allow us to find the recurrence relation determining the bulk geometry.…”

Section: Introductionmentioning

confidence: 99%

Dual Geometry of Entanglement Entropy via Deep Learning

Park¹,

Hwang²,

Cho³

et al. 2022

Preprint

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For a given entanglement entropy of QFT, we investigate how to reconstruct its dual geometry by applying the Ryu-Takayanagi formula and the deep learning method. In the holographic setup, the radial direction of the dual geometry is identified with the energy scale of the dual QFT. Therefore, the holographic dual geometry can describe how the QFT changes along the RG flow. Intriguingly, we show that the reconstructed geometry only from the entanglement entropy data can give us more information about other physical properties like thermodynamic quantities in the IR region.

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Machine Learning for Hilbert Series

Abstract: Hilbert series are a standard tool in algebraic geometry, and more recently are finding many uses in theoretical physics. This summary reviews work applying machine learning to databases of them; and was prepared for the proceedings of the Nankai Symposium on Mathematical Dialogues, 2021.

Cited by 3 publications

References 58 publications

Machine Learning Algebraic Geometry for Physics

Machine Learning Algebraic Geometry for Physics

Cluster Algebras: Network Science and Machine Learning

Dual Geometry of Entanglement Entropy via Deep Learning

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