Entropy 2013, 15(9), 3714-3733; doi:10.3390/e15093714
Article

Consideration on Singularities in Learning Theory and the Learning Coefficient

Department of Mathematics, College of Science & Technology, Nihon University, 1-8-14, Surugadai, Kanda, Chiyoda-ku, Tokyo 101-8308, Japan
Received: 21 June 2013; in revised form: 29 August 2013 / Accepted: 30 August 2013 / Published: 6 September 2013
(This article belongs to the Special Issue The Information Bottleneck Method)
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Abstract: We consider the learning coefficients in learning theory and give two new methods for obtaining these coefficients in a homogeneous case: a method for finding a deepest singular point and a method to add variables. In application to Vandermonde matrix-type singularities, we show that these methods are effective. The learning coefficient of the generalization error in Bayesian estimation serves to measure the learning efficiency in singular learning models. Mathematically, the learning coefficient corresponds to a real log canonical threshold of singularities for the Kullback functions (relative entropy) in learning theory.
Keywords: learning coefficient; Kullback function (relative entropy); singular learning machine; resolution of singularities

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MDPI and ACS Style

Aoyagi, M. Consideration on Singularities in Learning Theory and the Learning Coefficient. Entropy 2013, 15, 3714-3733.

AMA Style

Aoyagi M. Consideration on Singularities in Learning Theory and the Learning Coefficient. Entropy. 2013; 15(9):3714-3733.

Chicago/Turabian Style

Aoyagi, Miki. 2013. "Consideration on Singularities in Learning Theory and the Learning Coefficient." Entropy 15, no. 9: 3714-3733.

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