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Two birational invariants in statistical learning theory. (English) Zbl 1317.62005

Blanlœil, Vincent (ed.) et al., Singularities in geometry and topology, Strasbourg 2009. Proceedings of the 5th Franco-Japanese symposium on singularities, Strasbourg, France, August 24–28, 2009. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-118-7/pbk). IRMA Lectures in Mathematics and Theoretical Physics 20, 249-268 (2012).
Summary: We introduce 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.
For the entire collection see [Zbl 1255.14001].

MSC:

62B10 Statistical aspects of information-theoretic topics
60F05 Central limit and other weak theorems
13P25 Applications of commutative algebra (e.g., to statistics, control theory, optimization, etc.)
14E05 Rational and birational maps
62F99 Parametric inference
65B10 Numerical summation of series
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