Equations of States in Singular Statistical Estimation

H. Hironaka, “Resolution of singularities of an algebraic variety over a field of characteristic zero,” Ann. of Math., Vol.79, 109-326,1964

1964

Earlier work this paper cites.

M.F.Atiyah,“Resolution of singularities and division of distributions,” Comm. Pure Appl. Math., Vol.13,pp.145-150,1970

1970

Earlier work this paper cites.

M. Kashiwara, “B-functions and holonomic systems,” Inventions Math., 38, 33-53.1976

1976

Earlier work this paper cites.

J.A.Hartigan,“A failure of likelihood asymptotics for normal mixture,” Proc. of Barkeley Conf. in honor of Jerzy Neyman and Jack Keifer, Vol.2, pp.807-810,1985

1985

Earlier work this paper cites.

S.Watanabe, “Generalized Bayesian framework for neural networks with singular Fisher information matrices,” Proc. of International Symposium on Nonlinear Theory and Its applications, (Las Vegas), pp.207-210, 1995

1995

Earlier work this paper cites.

A. W. van der Vaart, Jon A. Wellner, “Weak Convergence and Empirical Processes,” Springer,1996

1996

Earlier work this paper cites.

S.Watanabe, “Algebraic analysis for singular statistical estimation,” Proc. of International Journal of Algorithmic Learning Theory, Lecture Notes on Computer Sciences, 1720, pp.39-50, 1999

1999

Earlier work this paper cites.

S. Watanabe, gAlgebraic information geometry for learning machines with singularities,h Advances in Neural Information Processing Systems, (Denver, USA), pp.329-336. 2001

2001

Earlier work this paper cites.

S.Watanabe, ”Algebraic analysis for nonidentifiable learning machines,” Neural Computation, Vol.13, No.4, pp.899-933, 2001

2001

Earlier work this paper cites.

S. Watanabe, ”Algebraic geometrical methods for hierarchical learning machines,” Neural Networks, Vol.14, No.8,pp.1049-1060, 2001

2001

Cited alongside, same era.

S. Watanabe, ”Learning efficiency of redundant neural networks in Bayesian estimation,” IEEE Transactions on Neural Networks, Vol.12, No.6, 1475-1486, 2001

2001

Cited alongside, same era.

K. Hagiwara, “On the Problem in Model Selection of Neural Network Regression in Overrealizable Scenario, ” Neural Comput., Vol.14,Vol.8, pp.1979 - 2002, 2002

2002

Cited alongside, same era.

S.Watanabe, S.-I.Amari,”Learning coefficients of layered models when the true distribution mismatches the singularities”, Neural Computation, Vol.15,No.5,1013-1033, 2003

2003

Cited alongside, same era.

K.Yamazaki, S.Watanabe,“Singularities in mixture models and upper bounds of stochastic complexity.” International Journal of Neural Networks, Vol.16, No.7, pp.1029-1038,2003

K.Yamazaki, S.Watanabe,“ Singularities in Complete bipartite graph-type Boltzmann machines and upper bounds of stochastic complexities”, IEEE Trans. on Neural Networks, Vol. 16 (2), pp.312-324, 2005

2005

Later among the works it cites.

K. Yamazaki and S. Watanabe, ”Algebraic geometry and stochastic complexity of hidden Markov models”, Neurocomputing, Vol.69, pp.62-84, 2005

2005

Later among the works it cites.

S.-i. Amari, H. Park, and T. Ozeki, “Singularities Affect Dynamics of Learning in Neuromanifolds,” Neural Comput., 18(5),pp.1007 - 1065,2006

2006

Later among the works it cites.

K.Nagata, S.Watanabe, ”The Exchange Monte Carlo Method for Bayesian Learning in Singular Learning Machines,” Proc. of WCCI2006,(Canada, Cancouver), 2006

2006

Later among the works it cites.

S. Watanabe,“Algebraic geometry and learning theory,h Morikita publishing, 2006

2006