2006 Fiscal Year Final Research Report Summary
Establishment of Algebraic Geometrical Methods in Statistical Learning Theory
| Project/Area Number |
15500130
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Sensitivity informatics/Soft computing
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| Research Institution | Tokyo Institute of Technology |
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Principal Investigator |
WATANABE Sumio Tokyo Institute of Technology, P&I Lab, Professor, 精密工学研究所, 教授 (80273118)
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| Project Period (FY) |
2003 – 2006
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| Keywords | Algebraic Geometry / Statistical Learning Theory / Zeta function / Learning System / Learning Machine / Learning Theory / Algebraic Analysis / Information System |
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| Research Abstract |
A lot of learning machines such as artificial neural networks, normal mixtures, hidden Markov models, Boltzmann machines, Bayesian networks, reduced rank regressions are nonidentifiable and singular learning machines. They have degenerate Fisher Information matrices resulting that the conventional statistical learning theory does not hold. In order to establish the new learning theory, we need algebraic geometrical method to treat the likelihood functions. In this research, we have the following results. (1) The fundamental form of the likelihood function were introduced based on desingularization problem. (2) The asymptotic form of the marginal likelihood were derived based on the zeta function of learning theory. (3) The asymptotic behavior of the variational Bayes or the mean field approximation were obtained. (4) The asymptotic behavior of the empirical Bayes appriximation were obtained. These are the mathematical foundation originally established in this research.
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