메뉴 건너뛰기




Volumn 18, Issue 5, 2006, Pages 1007-1065

Singularities affect dynamics of learning in neuromanifolds

Author keywords

[No Author keywords available]

Indexed keywords


EID: 33646407804     PISSN: 08997667     EISSN: 1530888X     Source Type: Journal    
DOI: 10.1162/neco.2006.18.5.1007     Document Type: Article
Times cited : (109)

References (56)
  • 1
    • 0034202524 scopus 로고    scopus 로고
    • Nonmonotonic generalization bias of gaussian mixture models
    • Akaho, S., & Kappen, H. J. (2000). Nonmonotonic generalization bias of gaussian mixture models. Neural Computation, 12, 6, 1411-1428.
    • (2000) Neural Computation , vol.12 , Issue.6 , pp. 1411-1428
    • Akaho, S.1    Kappen, H.J.2
  • 2
    • 0016355478 scopus 로고
    • A new look at the statistical model identification
    • Akaike, H. (1974). A new look at the statistical model identification. IEEE Trans. Appl. Comp., AC-19, 716-723.
    • (1974) IEEE Trans. Appl. Comp. , vol.AC-19 , pp. 716-723
    • Akaike, H.1
  • 3
    • 0000920843 scopus 로고
    • Theory of adaptive pattern classifiers
    • Amari, S. (1967). Theory of adaptive pattern classifiers. IEEE Trans., EC-16(3), 299-307.
    • (1967) IEEE Trans. , vol.EC-16 , Issue.3 , pp. 299-307
    • Amari, S.1
  • 4
    • 0017713690 scopus 로고
    • Dynamics of pattern formation in lateral-inhibition type neural fields
    • Amari, S. (1977). Dynamics of pattern formation in lateral-inhibition type neural fields. Biological Cybernetics. 27, 77-87.
    • (1977) Biological Cybernetics , vol.27 , pp. 77-87
    • Amari, S.1
  • 5
    • 0002975366 scopus 로고
    • Differential geometry of a parametric family of invertible linear systems - Riemannian metric, dual affine connections and divergence
    • Amari, S. (1987). Differential geometry of a parametric family of invertible linear systems - Riemannian metric, dual affine connections and divergence. Mathematical Systems Theory, 20, 53-82.
    • (1987) Mathematical Systems Theory , vol.20 , pp. 53-82
    • Amari, S.1
  • 6
    • 0000396062 scopus 로고    scopus 로고
    • Natural gradient works efficiently in learning
    • Amari, S. (1998). Natural gradient works efficiently in learning. Neural Computation, 10, 251-276.
    • (1998) Neural Computation , vol.10 , pp. 251-276
    • Amari, S.1
  • 9
    • 0000729504 scopus 로고
    • Statistical theory of learning curves under entropic loss criterion
    • Amari, S., & Murata, N. (1993). Statistical theory of learning curves under entropic loss criterion. Neural Computation, 5, 140-154.
    • (1993) Neural Computation , vol.5 , pp. 140-154
    • Amari, S.1    Murata, N.2
  • 11
    • 17144406556 scopus 로고    scopus 로고
    • Difficulty of singularity in population coding
    • Amari, S., & Nakahara, H. (2005). Difficulty of singularity in population coding. Neural Computation, 17, 839-858.
    • (2005) Neural Computation , vol.17 , pp. 839-858
    • Amari, S.1    Nakahara, H.2
  • 12
    • 0035125794 scopus 로고    scopus 로고
    • Differential and algebraic geometry of multilayer perceptrons
    • Amari, S., & Ozeki, T. (2001). Differential and algebraic geometry of multilayer perceptrons. IEICE Trans., E84-A, 31-38.
    • (2001) IEICE Trans. , vol.E84-A , pp. 31-38
    • Amari, S.1    Ozeki, T.2
  • 13
    • 0038030649 scopus 로고    scopus 로고
    • Learning and inference in hierarchical models with singularities
    • Amari, S., Ozeki, T., & Park, H. (2003). Learning and inference in hierarchical models with singularities. Systems and Computers in Japan, 34, 34-42.
    • (2003) Systems and Computers in Japan , vol.34 , pp. 34-42
    • Amari, S.1    Ozeki, T.2    Park, H.3
  • 14
    • 0034201611 scopus 로고    scopus 로고
    • Adaptive method of realizing natural gradient learning for multilayer perceptrons
    • Amari, S., Park, H., & Fukumizu, K. (2000). Adaptive method of realizing natural gradient learning for multilayer perceptrons. Neural Computation, 12, 1399-1409.
    • (2000) Neural Computation , vol.12 , pp. 1399-1409
    • Amari, S.1    Park, H.2    Fukumizu, K.3
  • 15
    • 33646435394 scopus 로고    scopus 로고
    • Statistical inference in nonidentifiable and singular statistical models
    • Amari, S., Park, H., & Ozeki, T. (2001). Statistical inference in nonidentifiable and singular statistical models. J. of the Korean Statistical Society, 30(2), 179-192.
    • (2001) J. of the Korean Statistical Society , vol.30 , Issue.2 , pp. 179-192
    • Amari, S.1    Park, H.2    Ozeki, T.3
  • 16
    • 84898934178 scopus 로고    scopus 로고
    • Geometrical singularities in the neuromanifold of multilayer perceptrons
    • T. G. Dietterich, S. Becker, & Z. Ghahramani (Eds.). Cambridge, MA: MIT Press
    • Amari, S., Park, H., & Ozeki, T. (2002). Geometrical singularities in the neuromanifold of multilayer perceptrons. In T. G. Dietterich, S. Becker, & Z. Ghahramani (Eds.), Advances in neural information processing systems, 14 (pp. 343-350). Cambridge, MA: MIT Press.
    • (2002) Advances in Neural Information Processing Systems , vol.14 , pp. 343-350
    • Amari, S.1    Park, H.2    Ozeki, T.3
  • 17
    • 0016991025 scopus 로고
    • Some geometric questions in the theory of linear systems
    • Brockett, R. W. (1976). Some geometric questions in the theory of linear systems. IEEE Trans. on Automatic Control, 21, 449-455.
    • (1976) IEEE Trans. on Automatic Control , vol.21 , pp. 449-455
    • Brockett, R.W.1
  • 18
    • 0001199897 scopus 로고
    • On the geometry of feedforward neural network error surfaces
    • Chen, A. M., Lu, H., & Hecht-Nielsen, R. (1993). On the geometry of feedforward neural network error surfaces. Neural Computation, 5, 910-927.
    • (1993) Neural Computation , vol.5 , pp. 910-927
    • Chen, A.M.1    Lu, H.2    Hecht-Nielsen, R.3
  • 19
    • 84996128633 scopus 로고    scopus 로고
    • Testing in locally conic models, and application to mixture models
    • Dacunha-Castelle, D., & Gassiat, É. (1997). Testing in locally conic models, and application to mixture models. Probability and Statistics, 1, 285-317.
    • (1997) Probability and Statistics , vol.1 , pp. 285-317
    • Dacunha-Castelle, D.1    Gassiat, É.2
  • 21
    • 0042311289 scopus 로고    scopus 로고
    • Likelihood ratio of unidentifiable models and multilayer neural networks
    • Fukumizu, K. (2003). Likelihood ratio of unidentifiable models and multilayer neural networks. Annals of Statistics, 32(3), 833-851.
    • (2003) Annals of Statistics , vol.32 , Issue.3 , pp. 833-851
    • Fukumizu, K.1
  • 22
    • 0034018074 scopus 로고    scopus 로고
    • Local minima and plateaus in hierarchical structures of multilayer perceptrons
    • Fukumizu, K., & Amari, S. (2000), Local minima and plateaus in hierarchical structures of multilayer perceptrons. Neural Networks, 13, 317-327.
    • (2000) Neural Networks , vol.13 , pp. 317-327
    • Fukumizu, K.1    Amari, S.2
  • 23
    • 0040079861 scopus 로고    scopus 로고
    • On the problem in model selection of neural network regression in overrealizable scenario
    • Hagiwara, K. (2002a). On the problem in model selection of neural network regression in overrealizable scenario. Neural Computation, 14, 1979-2002.
    • (2002) Neural Computation , vol.14 , pp. 1979-2002
    • Hagiwara, K.1
  • 24
    • 0036826115 scopus 로고    scopus 로고
    • Regularization learning, early stopping and biased estimator
    • Hagiwara, K. (2002b). Regularization learning, early stopping and biased estimator. Neurocomputing, 48, 937-955.
    • (2002) Neurocomputing , vol.48 , pp. 937-955
    • Hagiwara, K.1
  • 25
    • 0035200563 scopus 로고    scopus 로고
    • Upper bound of the expected training error of neural network regression for a gaussian noise sequence
    • Hagiwara, K., Hayasaka, T., Toda, N., Usui, S., & Kuno, K. (2001). Upper bound of the expected training error of neural network regression for a gaussian noise sequence. Neural Networks, 14, 1419-1429.
    • (2001) Neural Networks , vol.14 , pp. 1419-1429
    • Hagiwara, K.1    Hayasaka, T.2    Toda, N.3    Usui, S.4    Kuno, K.5
  • 26
    • 0027884625 scopus 로고
    • On the problem of applying AIC to determine the structure of a layered feed-forward neural network
    • Nagoya, Japan
    • Hagiwara, K., Toda, N., & Usui, S. (1993). On the problem of applying AIC to determine the structure of a layered feed-forward neural network. Proceedings of IJCNN (Vol. 3, pp. 2263-2266). Nagoya, Japan.
    • (1993) Proceedings of IJCNN , vol.3 , pp. 2263-2266
    • Hagiwara, K.1    Toda, N.2    Usui, S.3
  • 28
    • 0001070336 scopus 로고
    • Tubes and spheres in n-spaces, and a class of statistical problems
    • Hotelling, H. (1939). Tubes and spheres in n-spaces, and a class of statistical problems. Amer. J. Math., 61, 440-460.
    • (1939) Amer. J. Math. , vol.61 , pp. 440-460
    • Hotelling, H.1
  • 29
    • 0038323312 scopus 로고    scopus 로고
    • On-line learning theory of soft committee machines with correlated hidden units - Steepest gradient descent and natural gradient descent
    • Inoue, M., Park, H., & Okada, M. (2003). On-line learning theory of soft committee machines with correlated hidden units - Steepest gradient descent and natural gradient descent. J. Phys. Soc. Jpn., 72(4), 805-810.
    • (2003) J. Phys. Soc. Jpn. , vol.72 , Issue.4 , pp. 805-810
    • Inoue, M.1    Park, H.2    Okada, M.3
  • 30
    • 0040899705 scopus 로고
    • Generalization in a two-layer neural networks
    • Kang, K., Oh, J.-H., Kwon, S., & Park, Y. (1993). Generalization in a two-layer neural networks. Phys. Rev. E, 48(6), 4805-4809.
    • (1993) Phys. Rev. E , vol.48 , Issue.6 , pp. 4805-4809
    • Kang, K.1    Oh, J.-H.2    Kwon, S.3    Park, Y.4
  • 31
    • 20344383126 scopus 로고    scopus 로고
    • On the statistical properties of least squares estimators of layered neural networks (in Japanese)
    • Kitahara, M., Hayasaka, T., Toda, N., & Usui, S. (2000). On the statistical properties of least squares estimators of layered neural networks (in Japanese). IEICE Transactions, J86-D-II, 563-570.
    • (2000) IEICE Transactions, J86-D-II , pp. 563-570
    • Kitahara, M.1    Hayasaka, T.2    Toda, N.3    Usui, S.4
  • 32
    • 0000720909 scopus 로고
    • Functionally equivalent feedforward neural networks
    • Kůrková, V., & Kainen, P. C. (1994). Functionally equivalent feedforward neural networks. Neural Computation, 6, 543-558.
    • (1994) Neural Computation , vol.6 , pp. 543-558
    • Kůrková, V.1    Kainen, P.C.2
  • 33
    • 0041810454 scopus 로고    scopus 로고
    • Asymptotics for likelihood ratio tests under loss of identifiability
    • Liu, X., & Shao, Y. (2003). Asymptotics for likelihood ratio tests under loss of identifiability. Annals of Statistics, 32(3), 807-832.
    • (2003) Annals of Statistics , vol.32 , Issue.3 , pp. 807-832
    • Liu, X.1    Shao, Y.2
  • 35
    • 0028544395 scopus 로고
    • Network information criterion -determining the number of hidden units for an artificial neural network model
    • Murata, N., Yoshizawa, S., & Amari, S. (1994). Network information criterion -determining the number of hidden units for an artificial neural network model. IEEE Transactions on Neural Networks, 5(6), 865-872.
    • (1994) IEEE Transactions on Neural Networks , vol.5 , Issue.6 , pp. 865-872
    • Murata, N.1    Yoshizawa, S.2    Amari, S.3
  • 36
    • 0034266869 scopus 로고    scopus 로고
    • Adaptive natural gradient learning algorithms for various stochastic models
    • Park, H., Amari, S., & Fukumizu, K. (2000). Adaptive natural gradient learning algorithms for various stochastic models. Neural Networks, 13, 755-764.
    • (2000) Neural Networks , vol.13 , pp. 755-764
    • Park, H.1    Amari, S.2    Fukumizu, K.3
  • 37
    • 0345581643 scopus 로고    scopus 로고
    • On-line learning dynamics of multilayer perceptrons with unidentifiable parameters
    • Submitted to
    • Park, H., Inoue, M., & Okada, M. (2003). On-line learning dynamics of multilayer perceptrons with unidentifiable parameters. Submitted to J. Phys. A: Math. Gen., 36, 11753-11764.
    • (2003) J. Phys. A: Math. Gen. , vol.36 , pp. 11753-11764
    • Park, H.1    Inoue, M.2    Okada, M.3
  • 38
    • 0032613639 scopus 로고    scopus 로고
    • Analysis of natural gradient descent for multilayer neural networks
    • Rattray, M., & Saad, D. (1999). Analysis of natural gradient descent for multilayer neural networks. Physical Review E, 59, 4523-4532.
    • (1999) Physical Review E , vol.59 , pp. 4523-4532
    • Rattray, M.1    Saad, D.2
  • 39
    • 0000198852 scopus 로고    scopus 로고
    • Natural gradient descent for on-line learning
    • Rattray, M., Saad, D., & Amari, S. (1998). Natural gradient descent for on-line learning. Physical Review Letters, 81, 5461-5464.
    • (1998) Physical Review Letters , vol.81 , pp. 5461-5464
    • Rattray, M.1    Saad, D.2    Amari, S.3
  • 40
    • 21844501969 scopus 로고
    • On-line backpropagation in two-layered neural networks
    • Riegler, P., & Biehl, M. (1995). On-line backpropagation in two-layered neural networks. J. Phys. A; Math. Gen., 28, L507-L513.
    • (1995) J. Phys. A; Math. Gen. , vol.28
    • Riegler, P.1    Biehl, M.2
  • 41
    • 0000318553 scopus 로고
    • Stochastic complexity and modeling
    • Risssanen, J. (1986). Stochastic complexity and modeling. Ann. Statist. 14, 1080-1100.
    • (1986) Ann. Statist. , vol.14 , pp. 1080-1100
    • Risssanen, J.1
  • 45
    • 4243234689 scopus 로고
    • On-line learning in soft committee machines
    • Saad, D., & Solla, A. (1995). On-line learning in soft committee machines. Phys. Rev. E, 52, 4225-4243.
    • (1995) Phys. Rev. E , vol.52 , pp. 4225-4243
    • Saad, D.1    Solla, A.2
  • 46
    • 0000120766 scopus 로고
    • Estimating the dimension of a model
    • Schwarz, G. (1978). Estimating the dimension of a model. Annals of Statistics, 6, 461-164.
    • (1978) Annals of Statistics , vol.6 , pp. 461-1164
    • Schwarz, G.1
  • 47
    • 0026897370 scopus 로고
    • Uniqueness of the weights for minimal feedforward nets with a given input-output map
    • Sussmann, H. J. (1992). Uniqueness of the weights for minimal feedforward nets with a given input-output map. Neural Networks. 5, 589-593.
    • (1992) Neural Networks. , vol.5 , pp. 589-593
    • Sussmann, H.J.1
  • 48
    • 0035316373 scopus 로고    scopus 로고
    • Algebraic analysis for non-identifiable learning machines
    • Watanabe, S. (2001a). Algebraic analysis for non-identifiable learning machines. Neural Computation, 23, 899-933.
    • (2001) Neural Computation , vol.23 , pp. 899-933
    • Watanabe, S.1
  • 49
    • 0034842476 scopus 로고    scopus 로고
    • Algebraic geometrical methods for hierarchical learning machines
    • Watanabe, S. (2001b). Algebraic geometrical methods for hierarchical learning machines. Neural Networks, 24(8), 1409-1060.
    • (2001) Neural Networks , vol.24 , Issue.8 , pp. 1409-11060
    • Watanabe, S.1
  • 50
    • 0003937573 scopus 로고    scopus 로고
    • Algebraic information geometry for learning machines with singularities
    • T. K. Leen, T. G. Dietterich, & V. Tresp (Eds.). Cambridge, MA: MIT Press
    • Watanabe, S. (2001c). Algebraic information geometry for learning machines with singularities. In T. K. Leen, T. G. Dietterich, & V. Tresp (Eds.), Advances in neural information processing systems, 13 (pp. 329-336). Cambridge, MA: MIT Press.
    • (2001) Advances in Neural Information Processing Systems , vol.13 , pp. 329-336
    • Watanabe, S.1
  • 51
    • 0042879444 scopus 로고    scopus 로고
    • Learning coefficients of layered models when the true distribution mismatches the singularities
    • Watanabe, S., & Amari, S. (2003). Learning coefficients of layered models when the true distribution mismatches the singularities. Neural Computation, 15(5), 1013-1033.
    • (2003) Neural Computation , vol.15 , Issue.5 , pp. 1013-1033
    • Watanabe, S.1    Amari, S.2
  • 52
    • 0000045032 scopus 로고
    • On the volume of tubes
    • Weyl, H. (1939). On the volume of tubes. Amer. J. Math., 61, 461-472.
    • (1939) Amer. J. Math. , vol.61 , pp. 461-472
    • Weyl, H.1
  • 53
    • 0036583146 scopus 로고    scopus 로고
    • Population coding and decoding in a neural field: A computational study
    • Wu, S., Amari, S., & Nakahara, H. (2002). Population coding and decoding in a neural field: A computational study. Neural Computation, 14, 999-1026.
    • (2002) Neural Computation , vol.14 , pp. 999-1026
    • Wu, S.1    Amari, S.2    Nakahara, H.3
  • 54
    • 0035319119 scopus 로고    scopus 로고
    • Population coding with correlation and an unfaithful model
    • Wu, S., Nakahara, H., & Amari, S. (2001). Population coding with correlation and an unfaithful model. Neural Computation, 23, 775-797.
    • (2001) Neural Computation , vol.23 , pp. 775-797
    • Wu, S.1    Nakahara, H.2    Amari, S.3
  • 55
    • 0742318729 scopus 로고    scopus 로고
    • A probabilistic algorithm to calculate the learning curves of hierarchical learning machines with singularities
    • Yamazaki, K., & Watanabe, S. (2002). A probabilistic algorithm to calculate the learning curves of hierarchical learning machines with singularities. Trans. on IEICE, J85-D-II(3), 363-372.
    • (2002) Trans. on IEICE, J85-D-II , Issue.3 , pp. 363-372
    • Yamazaki, K.1    Watanabe, S.2
  • 56
    • 0742324924 scopus 로고    scopus 로고
    • Singularities in mixture models and upper bounds of stochastic complexity
    • Yamazaki, K., & Watanabe, S. (2003). Singularities in mixture models and upper bounds of stochastic complexity. International Journal of Neural Networks, 16(7), 1029-1038.
    • (2003) International Journal of Neural Networks , vol.16 , Issue.7 , pp. 1029-1038
    • Yamazaki, K.1    Watanabe, S.2


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.