Algebraic Analysis for Singular Statistical Estimation

  • Sumio Watanabe
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1720)


This paper clarifies learning efficiency of a non-regular parametric model such as a neural network whose true parameter set is an analytic variety with singular points. By using Sato’s b-function we rigorously prove that the free energy or the Bayesian stochastic complexity is asymptotically equal to λ 1 log n − (m 1 − 1) log log n+constant, where λ 1 is a rational number, m 1 is a natural number, and n is the number of training samples. Also we show an algorithm to calculate λ 1 and m 1 based on the resolution of singularity. In regular models, 2λ 1 is equal to the number of parameters and m 1 = 1, whereas in non-regular models such as neural networks, 2λ 1 is smaller than the number of parameters and m 1 ≥ 1.


Generalization Error Regular Model Algebraic Analysis Layered Neural Network Statistical Estimation Error 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Sumio Watanabe
    • 1
  1. 1.P&I Lab.Tokyo Institute of TechnologyYokohamaJapan

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