# Two birational invariants in statistical learning theory

### Sumio Watanabe

Tokyo Institute of Technology, Yokohama, Japan

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## Abstract

This paper introduces a recent advance in the research between algebraic geometry and statistical learning theory. A lot of statistical models used in information science contain singularities in their parameter spaces, to which the conventional theory can not be applied. The statistical foundation of singular models was been left unknown, because no mathematical base could be found. However, recently new theory was constructed based on algebraic geometry and algebraic analysis. In this paper, we show that statistical estimation process is determined by two birational invariants, the real log canonical threshold and the singular fluctuation. As a result, a new formula is derived, which enables us to estimate the generalization error without any knowledge of the information source. In the discussion, a relation between mathematics and the real world is introduced to pure mathematicians.