Behavior of Large random tensors and related topics

2023 Fiscal Year Final Research Report

• Project/Area Number: 20K20882
• Research Category: Grant-in-Aid for Challenging Research (Exploratory)
• Allocation Type: Multi-year Fund
• Review Section: Medium-sized Section 12:Analysis, applied mathematics, and related fields
• Research Institution: Kyoto University
• Principal Investigator: Collins Benoit 京都大学, 理学研究科, 教授 (20721418)
• Project Period (FY): 2020-07-30 – 2024-03-31
• Keywords: random tenors / free probability

Outline of Final Research Achievements

In the fiscal year 2020, research on extending the HCIZ integral to the tensor setting was published in JEMS, and research on applying free probability theory to neural networks was published in CMP. In the fiscal year 2021, research on the eigenvalue distribution of the sum of k-th order tensor products of rank-1 independent n-dimensional vectors was published in EJP. In the fiscal year 2022, research on the asymptotic behavior when the size N of the tensor is large, considering the generalization of the HCIZ integral to tensors, was published in CMP, and research on the spectrum of local random Hamiltonians was published in JPhA. In the fiscal year 2023, a paper demonstrating the asymptotic freeness of unitary matrices in tensor product spaces for invariant states was published in RMTA.

Free Research Field

random tensors

Academic Significance and Societal Importance of the Research Achievements

ランダムテンソルの研究は、2本の脚を持つランダム行列に比べて非常に限られていたが本研究課題により、その手法を大きく広げたり、新しい現象を発見した。また機械学習に関連する研究も生まれた。量子情報理論や量子重力理論でもランダムテンソルが必要とされる場面があり、機械学習理論の進歩により、データ構造にテンソル構造が含まれるケースが増えています。これにより、ランダムテンソルの研究が進展しています。この研究成果は、今後学術的には新しい数学的ツールの開発を促進し、社会的には量子情報理論や機械学習の進歩に貢献する可能性があります。