2021 Fiscal Year Annual Research Report
Machine learning and statistical methhods on infinite-dimensional manifolds
| Project/Area Number |
20H04250
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| Research Institution | Institute of Physical and Chemical Research |
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Principal Investigator |
Ha QuangMinh 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 | covariance operators / Gaussian measures / Gaussian processes / optimal transport |
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| Outline of Annual Research Achievements |
We have achieved the following:
1) In the context of infinite-dimensional optimal transport, we obtained the explicit formulas for the entropic regularized Wasserstein distance between infinite-dimensional Gaussian measures on Hilbert spaces, including in particular the RKHS setting. Our mathematical analysis demonstrates explicitly the many desirable theoretical properties of the entropic regularization formulation over the exact Wasserstein distance.
2) We carried out sample complexity analysis for the finite-dimensional approximations of exact and entropic Wasserstein distances between infinite-dimensional covariance operators associated with stochastic processes, in particular Gaussian processes. For the entropic formulation, the complexities are all dimension-independent.
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| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
We are making satisfactory progress on the geometry and statistical properties of infinite-dimensional covariance operators, Gaussian measures, and Gaussian processes in the setting of Optimal Transport. Results in the direction of Information Geometry are forthcoming.
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| Strategy for Future Research Activity |
We are currently working on
1) Sample complexity analysis for finite-dimensional approximations of distances
between infinite-dimensional covariance operators, Gaussian measures, and Gaussian processes in the setting of Information Geometry.
2) Applications of the above results, in particular in Functional Data Analysis.
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