2022 Fiscal Year Final Research Report
Machine learning and statistical methhods on infinite-dimensional manifolds
| Project/Area Number |
20H04250
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Review Section |
Basic Section 61030:Intelligent informatics-related
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| Research Institution | Institute of Physical and Chemical Research |
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Principal Investigator |
Ha Quang Minh 国立研究開発法人理化学研究所, 革新知能統合研究センター, ユニットリーダー (90868928)
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| Project Period (FY) |
2020-04-01 – 2023-03-31
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| Keywords | Gaussian measures / Gaussian processes / Optimal Transport / Information Geometry / Riemannian geometry / Divergences / Wasserstein distance / Entropic regularization |
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| Outline of Final Research Achievements |
We have obtained many results on the geometry of infinite-dimensional Gaussian measures, Gaussian processes, and infinite-dimensional positive definite operators in the framework of Optimal Transport and Information Geometry. These include
(1) Explicit mathematical formulas for many quantities of interest involved, including entropic regularized Wasserstein distance, regularized Kullback-Leibler and Renyi divergences, and regularized Fisher-Rao distance. These can readily be employed in algorithms in machine learning and statistics.
(2) Extensive theoretical analysis showing in particular dimension-independent sample complexities of the above regularized distances and divergences. These provide guarantees for the consistency of finite-dimensional methods to approximate them in practice. Moreover, we show explicitly that the regularized distances and divergences possess many favorable theoretical properties over exact ones, with implications for practical algorithms.
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| Free Research Field |
Mathematical foundations of machine learning
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| Academic Significance and Societal Importance of the Research Achievements |
Our results are the first in the setting of infinite-dimensional Gaussian measures and Gaussian processes. They (1) elucidate many theoretical properties of Optimal Transport; (2) have important consequences for the mathematical foundations of Gaussian process methods in machine learning.
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