2020 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 | Information geometry / Fisher-Rao metric / Hilbert manifold / Gaussian process / RKHS |
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| Outline of Annual Research Achievements |
We have achieved the following:
(i) We proved that the set of infinite-dimensional centered Gaussian measures on a Hilbert space, which are equivalent to a fixed Gaussian measure, admits a smooth Hilbert manifold structure. We then formulated the Fisher-Rao Riemannian metric on this infinite-dimensional manifold, with closed form expressions for many quantities of interest, including the Riemannian metric, curvature tensor, geodesic curve, and geodesic distance. (ii) We connected this formulation with the geometric framework of positive definite unitized Hilbert-Schmidt operators on a Hilbert space. In particular, we showed that the latter framework leads to the regularized Fisher-Rao distance between Gaussian measures on a Hilbert space. The regularization framework leads to dimension-independent sample complexities for this distance in the reproducing kernel Hilbert space and Gaussian process settings.
We have also carried out preliminary numerical results utilizing this framework in the Gaussian process setting.
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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
Most work in information geometry and optimal transport has focused on the finite-dimensional setting.
Mathematical techniques for the finite-dimensional setting generally do not generalize directly to the infinite-dimensional setting. Our results are the first in the literature in the setting of infinite-dimensional Gaussian measures. These results require substantially mathematical machinery and techniques, which would also be of interest in their own right.
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| Strategy for Future Research Activity |
The theoretical results are expected to form a part of the basis for the mathematical foundations of Gaussian process methods in machine learning. In particular, The application of this framework for Bayesian Neural Networks is currently being explored.
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