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Sovable Models of Finite Multi-resolution analysis

Research Project

Project/Area Number 09680362
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Intelligent informatics
Research InstitutionTokyo Institute of Technology

Principal Investigator

WATANABE Sumio  Precision and Intelligence Laboratory, Associate Professor, 精密工学研究所, 助教授 (80273118)

Co-Investigator(Kenkyū-buntansha) 渡邊 澄夫  東京工業大学, 精密工学研究所, 助教授 (80273118)
Project Period (FY) 1997 – 1999
Project Status Completed (Fiscal Year 1999)
Budget Amount *help
¥2,100,000 (Direct Cost: ¥2,100,000)
Fiscal Year 1999: ¥300,000 (Direct Cost: ¥300,000)
Fiscal Year 1998: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 1997: ¥900,000 (Direct Cost: ¥900,000)
KeywordsMulti-Resolution Analysis / Neural networks / Learning Theory / Function approximation / Statistical Estimation / b-function / Singularities / Hironaka's resolution / ウェーブレット / ニューラルネットワーク / 代数解析 / 自由エネルギー / 確率的複雑さ / 可解モデル / 汎関数測度
Research Abstract

Learning models with finite multi-resolution functions, for example, neural networks, gaussian mixtures, and finite wavelets, are often used in pattern recognition, robotic control, and time sequence prediction. However, their mathematical foundation has not been established because they are not linear or not regular models. In this research, we clarified the two aspects of their mathematical properties, (1) function approximation abilities, and (2) statistical estimation effciencies.
(1) It is well known that function approximation errors by their models depend on the functional topologies. In this research, we proposed a method to clarify the function approximation errors based on the assumption that the target functions are randomly taken from the function probability measures. Based on this assumption, we proved that the average function approximation errors are determined by the covariance of the coeficients of the functions, or the sparseness of the functions. This result shows a critrion whether the multi-resolution analysis is useful or not.
(2) In order to clarify the statistical estimation errors, we have shown that the stochastic complexity of the learning model is determined by the deepest singularities of the model, and we developed an algorithm to calculate the learning efficiency based on the Sato-Bernstein's b-function and Hironaka's resolution of singularities.
The problems for the future are to depelop a method to estimate the functional probability measure of the images and sounds and to establish the mathematical foundation of the maximum likelihood method.

Report

(4 results)
  • 1999 Annual Research Report   Final Research Report Summary
  • 1998 Annual Research Report
  • 1997 Annual Research Report
  • Research Products

    (42 results)

All Other

All Publications (42 results)

  • [Publications] 渡邊澄夫: "ベイズ法による階層型統計モデルの汎化誤差について"電子情報通信学会論文誌. J81-A. 1442-1452 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Sumio Watanabe: "Realizable approximation bounds for a solvable neural network"Approximation Theory. Vol. 1. 347-354 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Sumio Watanabe: "Inequalities of Generalization errors for layered neural networks in Bayesian learning"Proc. of Int. Conf. on Neural Information Processing. Vol. 1. 59-62 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Sumio Watanabe: "Approximation bounds for layered lerning machines and environmental probability measures"Proc. of Int. Conf. on Computational Intelligence and Neuroscience. Vol. 2. 135-138 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Sumio Watanabe: "Algebraic analysis for neural network learning"Proc. of IEEE Systems, Man, and Cybernetics. CD ROM. (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Sumio Watanabe: "Algebraic analysis for singular statistical estimation"Lecture Notes on Computer Science. 1720. 39-50 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Sumio Watanabe: "Algebraic analysis for non-regular learning machines"Neural Information Processing Systems. 12. 356-362 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] 1-22 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] 1-18 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] S. Watanabe and K. Fukumizu: "Algorithms and Architectures"Academic Press. 456 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] 渡邊澄夫: "データ学習アルゴリズム"共立出版. 150 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] S. Watanabe: "On the generalization error by a layered statistical model with Bayesian estimation"IEICE Trans.. Vol. J81-A, No. 10. 1442-1452 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] S. Watanabe: "Realizable approximation bounds for a solvable neural network"Approximation Theory, Vol. 1, Vaderbilt University Press. 347-354 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] S. Watanabe: "Inequalities of Generalization Errors for Layered Neural Networks in Nayesian Learning"Proc. of Int. Conf. on Neural Information Processing. 59-62 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] S. Watanabe: "Approximation bounds for layered learning machines and environmental probability measures"Proc. of Int. Conf. on Computational Intelligence and Neuraoscience. Vol. 2. 135-138 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] S. Watanabe: "Algebraic analysis for neural network learning"Proc. of IEEE Systems, Man Cybernetics. (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] S. Watanabe: "Algebraic analysis for singular statistical estimation"Lecture Notes on Computer Sciences. 1720. 39-50 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] S. Watanabe: "Algebraic analysis for non-regular learning machines"Neural Information Processing Systems. 12. 356-362 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] S. Watanabe and K. Fukumizu: "Algorithms and Architectures"Academic Press. (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] S. Watanabe: "Algorithms for learning from data"Kyoritsu-Shuppan. (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] 渡邊澄夫: "特異点解消と佐藤 b関数による神経回路網の学習効率計算法"日本神経回路学会第9回全国大会講演論文集. 9巻. 163-164 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] 渡邊澄夫: "逆問題と佐藤b関数"統計数理研究所研究会「逆問題とその周辺」. 第6巻(掲載予定). (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] Sumio Watanabe: "Algebraic analysis for neural network learning"Proceedings of International Symposium on IEEE Systems,Man,and Cybernietics. CD ROM. (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Sumio Watanabe: "Matahematical foundation for redundant statistical estimation"Proceedings of Internatinal Conterence on Stochastic Systems. (掲載予定). (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] Sumio Watanabe: "Algebraic analysis for singular statistica estimation"Proceedings of International Symposium on Algorithmic Learning Theory. 10巻. 39-50 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Sumio Watanabe: "Algebraic analysis for non-regular learning machines"Advances in Neural Information Processing Systems. 12巻. 356-362 (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] 渡邊澄夫: "有界な解析的集合を真の解に持つモデルの最尤推定の評価"電子情報通信学会技術報告(ニューロコンピューティング). (掲載予定). (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] 渡邊澄夫: "統計的推測における特異点の問題"応用数理. (掲載予定). (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] 渡邊澄夫: "ベイズ法による階層型統計モデルの推定誤差について" 電子情報通信学会誌. J81-A,10号. 1442-1552 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] 渡邊澄夫: "代数解析に基づく特異点を持つモデルの学習理論" 電子情報通信学会技術報告. NC98-64. 73-80 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] 渡邊澄夫: "非正規な統計的推定における佐藤のb関数の役割" 情報理論とその応用. 21巻1号. 367-370 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] 渡邊澄夫: "神経回路網のバイアスとバリアンス" 統計数理研究所研究リポート「逆問題とその周辺」. 5巻(掲載予定). (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] 渡邊澄夫: "Approximation Bounds for Layered Learning Machines and Environmental Probability Measures" Proceedings of the Third International Conference on Computational Intelligence and Neuroscience. 2巻. 135-138 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] 渡邊澄夫: "Inequalities of Generalization Errors for Layered Neural Networks in Bayesian Learning" Proceedings of Fifth International Conference on Neural Information Processing. 1巻. 59-62 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] 渡邊澄夫 福水健次: "Algorithms and Architectures" Academic Press, 450 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] 渡邊澄夫: "A finite wavelet decomposition method" Electronics and Communication in Japan. 80巻7号. 1-10 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] 渡邊澄夫: "局所自由エネルギーと予測誤差の関係について" 日本神経回路学会第8回全国大会講演論文集. 8巻. 180-181 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] 渡邊澄夫: "Realizable approximation bounds for a solvable neural network" Approximation Theory. 9巻(掲載予定). (1998)

    • Related Report
      1997 Annual Research Report
  • [Publications] 渡邊澄夫: "階層型モデルにおける知識獲得と予測向上の非両立性" 統計数理研究所共同研究リポート「逆問題とその周辺」. 4巻(掲載予定). (1998)

    • Related Report
      1997 Annual Research Report
  • [Publications] 渡邊澄夫: "大自由度認知機構における隠れ系の統計的役割について" 物性研究. 69巻(掲載予定). (1998)

    • Related Report
      1997 Annual Research Report
  • [Publications] 渡邊澄夫: "関数空間上の測度による関数近似系の平均評価について" 電子情報通信学会技術報告. NC97巻(掲載予定). (1998)

    • Related Report
      1997 Annual Research Report
  • [Publications] 渡邊澄夫, 福永健次: "Neural network systems techniques and applicaitons" Academic Press, 300 (1997)

    • Related Report
      1997 Annual Research Report

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Published: 1997-04-01   Modified: 2016-04-21  

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