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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| Co-Investigator(Kenkyū-buntansha) |
山崎 啓介 東京工業大学, 精密工学研究所, 助手 (60376936)
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| Project Period (FY) |
2003 – 2006
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| Project Status |
Completed (Fiscal Year 2006)
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| Budget Amount *help |
¥2,900,000 (Direct Cost: ¥2,900,000)
Fiscal Year 2006: ¥600,000 (Direct Cost: ¥600,000)
Fiscal Year 2005: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 2004: ¥600,000 (Direct Cost: ¥600,000)
Fiscal Year 2003: ¥800,000 (Direct Cost: ¥800,000)
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| Keywords | Algebraic Geometry / Statistical Learning Theory / Zeta function / Learning System / Learning Machine / Learning Theory / Algebraic Analysis / Information System / 特異モデル / 確率的複雑さ / ベイズ法 / 変分ベイズ法 / ゼータ函数 / 縮小ランク回帰 / 文脈自由文法 / 汎化誤差 / カルバック情報量 / ブローアップ / 特定不能 |
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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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Report
(5 results)
Research Products
(38 results)
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[Journal Article] 局所化ベイズ学習法2006
Author(s)
高松慎吾, 中島伸一, 渡辺澄夫
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Journal Title
電子情報通信学会論文誌 Vol.J89-D-II, no.10
Pages: 2260-2268
Description
「研究成果報告書概要(和文)」より
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[Book] 代数幾何と学習理論2006
Author(s)
渡辺澄夫
Total Pages
200
Publisher
森北出版
Description
「研究成果報告書概要(和文)」より
Related Report
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[Book] 学習システムの理論と実現2005
Author(s)
渡邊澄夫, 萩原克幸, 赤穂昭太郎, 本村陽一, 福水健次, 岡田真人, 青柳美輝
Total Pages
195
Publisher
森北出版
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