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学習理論における特異性とその代数幾何学的性質およびその応用

Research Project

Project/Area Number 16700218
Research Category

Grant-in-Aid for Young Scientists (B)

Allocation TypeSingle-year Grants
Research Field Sensitivity informatics/Soft computing
Research InstitutionTokyo Institute of Technology (2006)
Sophia University (2004-2005)

Principal Investigator

青柳 美輝  東京工業大学, 精密工学研究所, 研究員 (90338434)

Project Period (FY) 2004 – 2006
Project Status Completed (Fiscal Year 2006)
Budget Amount *help
¥2,700,000 (Direct Cost: ¥2,700,000)
Fiscal Year 2006: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2005: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2004: ¥1,100,000 (Direct Cost: ¥1,100,000)
Keywords確率的複雑さ / 特異モデル / 特異点解消 / ゼータ関数 / 階層モデル / ベイズ推定 / 特異的モデル / 学習モデルのゼータ関数
Research Abstract

学習システムの目的は,たとえば,画像認識,音声認識,遺伝子解析,データマイニンである.このような学習データは,正規分布に従っているような単純なものではなく,非常に複雑な構造をもっている.したがって,このような目的のために用いられる学習モデルは十分複雑なものを考えなくてはならない.現在,そのようなデータを扱える学習モデルとして,階層構造,内部構造を持つものが応用されている.これらのモデルは,パラメータを変えることできわめて多くの確率分布となりえることが証明されている.このようなモデルを特異モデルという.渡邊澄夫教授は,特異モデルに関して,ベイズ推測において重要な役割を果たす確率的複雑さが,特異点解消により構成された方法論と数理的に緊密な関係を持つことを明らかにした.しかし,実はこのような数理の確立にもかかわらず,計算を行うことは非常に困難であった.特異点解消自身の困難性や,特異点の非孤立性,ニュートン図形の退化などが理由である.今年度は,これらの困難性を個別のモデルに対して工夫を行いながら克服し,昨年度よりさらに一般化した3層ニューラルネットワークの場合について,確率的複雑さの厳密な漸近展開を与えた.さらにその方法を混合正規分布にも応用した.方法の概略は昨年度と同様recursive blowing-up及びトーリック多様体による特異点を帰納的に解消する方法である.この方法で上限のみしか得られていなかった学習モデルのゼータ関数の最大の極およびその位数の値を与え,汎化誤差の漸近展開を明らかにした.今後は,スピンモデルについても考察していきたい。

Report

(3 results)
  • 2006 Annual Research Report
  • 2005 Annual Research Report
  • 2004 Annual Research Report
  • Research Products

    (20 results)

All 2006 2005 2004 Other

All Journal Article (18 results) Book (2 results)

  • [Journal Article] Generalization Error of Three Layered Learning Model in Bayesian Estimation2006

    • Author(s)
      Miki Aoyagi, Sumio Watanabe
    • Journal Title

      Proceedings of the Second IASTED International Conference on Computational Intelligence 2

      Pages: 405-410

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Eye Direction by Stereo Image Processing using Corneal Reflection on an Iris2006

    • Author(s)
      Kumiko Tsuji, Miki Aoyagi
    • Journal Title

      Proceedings of the Second IASTED International Conference on Computational Intelligence 2

      Pages: 355-360

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Stochastic Complexities of Reduced Rank Regression in Bayesian Estimation2005

    • Author(s)
      Miki Aoyagi, Sumio Watanabe
    • Journal Title

      Neural Networks 18

      Pages: 924-933

    • Related Report
      2005 Annual Research Report
  • [Journal Article] 特異点解消とニューラルネットワークのベイズ推定における汎化誤差2005

    • Author(s)
      Miki Aoyagi, Sumio Watanabe
    • Journal Title

      電子情報通信学会和文論文誌DII J88.10

      Pages: 2112-2124

    • NAID

      110003224123

    • Related Report
      2005 Annual Research Report
  • [Journal Article] The maximum pole of the zeta function for learning theory2005

    • Author(s)
      Miki Aoyagi, Sumio Watanabe
    • Journal Title

      Finite of Infinite Dimensional Complex Analysis and Applications

      Pages: 17-24

    • Related Report
      2005 Annual Research Report
  • [Journal Article] A modified genetic algorithm for feature clustering2005

    • Author(s)
      Miki Aoyagi, Kumiko Tsuji
    • Journal Title

      Finite or Infinite Dimensional complex Analysis and Applications

      Pages: 9-16

    • Related Report
      2005 Annual Research Report
  • [Journal Article] The Generalization Error of Reduced Rank Regression in Bayesian Estimation2004

    • Author(s)
      Miki Aoyagi, Sumio Watanabe
    • Journal Title

      International Symposium on Information Theory and its Applications

      Pages: 1068-1073

    • Related Report
      2004 Annual Research Report
  • [Journal Article] A modified genetic algorithm for image segmentation based on feature clustering2004

    • Author(s)
      Miki Aoyagi, Kumiko Tsuji
    • Journal Title

      International Symposium on Information Theory and its Applications

      Pages: 1074-1079

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Stochastic Complexity and Newton Diagram2004

    • Author(s)
      Keisuke Yamazaki, Miki Aoyagi, Sumio Watanabe
    • Journal Title

      International Symposium on Information Theory and its Applications

      Pages: 105-110

    • Related Report
      2004 Annual Research Report
  • [Journal Article] The Generalization Error of Reduced Rank Regression Models and Resolution of Singularities2004

    • Author(s)
      Miki Aoyagi, Sumio Watanabe
    • Journal Title

      Technical report of IEICE, NC2004

      Pages: 13-18

    • NAID

      110003232648

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Kullback Information of Normal Mixture is not an Analytic Function2004

    • Author(s)
      Sumio Watanabe, Keisuke Yamazaki, Miki Aoyagi
    • Journal Title

      Technical report of IEICE, NC2004

      Pages: 41-46

    • NAID

      110003232653

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Resolution of Singularities and the Generalization Error with Bayesian Estimation for Layered Neural Network

    • Author(s)
      Miki Aoyagi, Sumio Watanabe
    • Journal Title

      Systems and Computers in Japan (In press)

    • NAID

      110003224123

    • Related Report
      2006 Annual Research Report
  • [Journal Article] The zeta function of learning theory and generalization error of three layered neural perceptron

    • Author(s)
      Miki Aoyagi
    • Journal Title

      数理解析研究所講究録,Recent Topics on Real and Complex Singularities (In press)

    • Related Report
      2006 Annual Research Report
  • [Journal Article] The zeta function for learning theory and resolution of Singularities

    • Author(s)
      Miki Aoyagi, Sumio Watanabe
    • Journal Title

      Proceedings of the 5th ISAAC Congress 5(In press)

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Eye direction by stereo image processing

    • Author(s)
      Kumiko Tsuji, Miki Aoyagi
    • Journal Title

      Proceedings of the 5th ISAAC Congress 5(In press)

    • Related Report
      2006 Annual Research Report
  • [Journal Article] The zeta function for learning theory and resolution of singularities

    • Author(s)
      Miki Aoyagi, Sumio Watanabe
    • Journal Title

      Proceedings of the 5th ISAAC Congress (印刷中)

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Eye direction by stereo image processing.

    • Author(s)
      Miki Aoyagi, Kumiko Tsuji
    • Journal Title

      Proceedings of the 5th ISAAC Congress, (印刷中)

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Stochastic Complexities of Reduced Rank Regression in Bayesian Estimation

    • Author(s)
      Miki Aoyagi, Sumio Watanabe
    • Journal Title

      Neural Network (To appear)

    • Related Report
      2004 Annual Research Report
  • [Book] 学習システムの理論と実現2005

    • Author(s)
      青柳美輝, 赤穂昭太郎, 岡田真人, 萩原克幸, 福水健次, 本村陽一, 渡辺澄夫
    • Total Pages
      195
    • Publisher
      森北出版
    • Related Report
      2005 Annual Research Report
  • [Book] 学習システムの理論と実現2005

    • Author(s)
      青柳美輝, 赤穂昭太郎, 岡田真人, 萩原克幸, 福水健次, 本村陽一, 渡辺澄夫
    • Publisher
      森北出版(印刷中)
    • Related Report
      2004 Annual Research Report

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Published: 2004-04-01   Modified: 2016-04-21  

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