Nonlinear Feature Extraction: Deterministic Neural Networks

  • Gustavo Deco
  • Dragan Obradovic
Part of the Perspectives in Neural Computing book series (PERSPECT.NEURAL)

Abstract

Independent feature extraction, i.e independent component analysis, was formulated in Chapter 4 as a search for an invertible, volume preserving map which statistically decorrelates the output components in the case of an arbitrary, possibly non-Gaussian input distribution. The same chapter discussed in detail the case where input-output maps are linear, i.e. matrices. The first extension to nonlinear transformation was carried out in Chapter 5 where stochastic neural networks were used to perform statistical decorrelation of Boolean outputs. This chapter further extends nonlinear independent feature extraction by introducing a very general class of nonlinear input-output deterministic maps whose architecture guarantees bijectivity and volume preservation. The criteria for evaluating statistical dependence are those defined in Chapter 4: the cumulant expansion method and the minimization of the mutual information among output components. Atick and Redlich [6.1] and especially the two papers of Redlich [6.2]–[6.3] use similar information theoretic concepts and reversible cellular automata architectures in order to define how nonlinear decorrelation can be performed. Taylor and Coombes [6.4] presented an extension of Oja’s learning rule for polynomial, i.e. higher order neural networks.

Keywords

Entropy Covariance Bromide Retina Assure 

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References

  1. [6.1]
    J. Atick and A. Redlich: What Does the Retina Know about Natural Scenes. Neural Computation, 4, 196–210, 1992.CrossRefGoogle Scholar
  2. [6.2]
    A. Redlich: Redundancy Reduction as a Strategy for Unsupervised Learning. Neural Computation, 5, 289–304, 1993.CrossRefGoogle Scholar
  3. [6.3]
    A. Redlich: Supervised Factorial Learning. Neural Computation, 5, 750–766, 1993.CrossRefGoogle Scholar
  4. [6.4]
    J. G. Taylor and S. Coombes: Learning Higher Order Correlations. Neural Networks, 6, 423–427,1993.CrossRefGoogle Scholar
  5. [6.5]
    G. Deco and Brauer G: Nonlinear Higher-Order Statistical Decorrelation by Volume-Conserving Neural Architectures. Neural Networks, 8, 525–535, 1995.CrossRefGoogle Scholar
  6. [6.6]
    G. Deco and B. Schürmann: Learning Time Series Evolution by Unsupervised Extraction of Correlations. Physical Review E, 51, 1780–1790, 1995.CrossRefGoogle Scholar
  7. [6.7]
    G. Deco, L. Parra and S. Miesbach: Redundancy Reduction with Information-Preserving Nonlinear Maps. Network: Computation in Neural Systems, 6, 61–72, 1995.MATHCrossRefGoogle Scholar
  8. [6.8]
    J. Rubner and P. Tavan: A Self-Organization Network for Principal-Component Analysis. Europhysics Letters, 10, 693–698, 1989.CrossRefGoogle Scholar
  9. [6.9]
    R. Durbin and D. Rumelhart: Product Units: A Computationally Powerful and Biologically Plausible Extension to Backpropagation Networks. Neural Computation, 1, 133–142, 1989.CrossRefGoogle Scholar
  10. [6.10]
    C. Beck and F. Schlögl: Thermodynamics of Chaotic Systems. Cambridge Nonlinear Science Series, University Press, Cambridge, 1993.Google Scholar
  11. [6.11]
    P. Grassberger and I. Procaccia. Characterization of Strange Attractors. Phys. Rev. Lett., 50, 346, 1983.MathSciNetCrossRefGoogle Scholar
  12. [6.12]
    J.P. Eckmann and D. Ruelle. Ergodic Theory of Chaos and Strange Attractors. Rev. Mod. Phys., 57, 617–656, 1985.MathSciNetCrossRefGoogle Scholar
  13. [6.13]
    J.P. Crutchfield and McNamara: Equations of Motion from Data Series. Complex Systems, 1, 417–452, 1987.MathSciNetMATHGoogle Scholar
  14. [6.14]
    H.D. I. Abarbanel, R. Brown and J.B. Kadtke: Prediction and System Identification in Chaotic Time Series with Broadband Fourier Spectra. Phys. Lett. A, 138, 401–408, 1989.CrossRefGoogle Scholar
  15. [6.15]
    H.D.I. Abarbanel, R. Brown and J.B. Kadtke: Prediction in Chaotic Nonlinear Systems: Methods for Time Series with Broadband Fourier Spectra. Phys. Rev. A, 41, 1782–1807, 1990.MathSciNetCrossRefGoogle Scholar
  16. [6.16]
    J. Farmer and J. Sidorowich. Predicting Chaotic Time Series. Phys. Rev. Letters, 59, 845, 1987.MathSciNetCrossRefGoogle Scholar
  17. [6.17]
    M. Casdagli: Nonlinear Prediction of Chaotic Time Series. Physica D, 35, 335–356, 1989.MathSciNetMATHCrossRefGoogle Scholar
  18. [6.18]
    A. Lapedes and R. Farber: Nonlinear Signal Processing Using Neural Networks: Prediction and System Modeling. Tech. Rep. n LA-UR-87-2662, Los Alamos National Laboratory, Los Alamos, NM, 1987.Google Scholar
  19. [6.19]
    A. Weigend, D. Rumelhart and B. Huberman: Back-Propagation, Weight Elimination and Time Series Prediction. In Connectionist Models, Proc. 1990, Touretzky, Elman, Sejnowski and Hinton eds., 105–116, 1990.Google Scholar
  20. [6.20]
    G. Deco and B. Schürmann: Recurrent Neural Networks Capture the Dynamical Invariance of Chaotic Time Series. IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, 77-A (11), 1840–1845, 1994.Google Scholar
  21. [6.21]
    W. Liebert and H.G. Schuster: Proper Choice of the Time Delay for the Analysis of Chaotic Time Series. Phys. Lett. A, 142, 107–111, 1989.MathSciNetCrossRefGoogle Scholar
  22. [6.22]
    W. Liebert, K. Pawelzik and H.G. Schuster: Optimal Embedding of Chaotic Attractors from Topological Considerations. Europhysics Lett., 14, 521–526, 1991.MathSciNetCrossRefGoogle Scholar
  23. [6.23]
    K. Pawelzik and H.G. Schuster: Unstable Periodic Orbits and Prediction. Phys. Rev. A, 43, 1808–1812, 1991.CrossRefGoogle Scholar
  24. [6.24]
    F. Takens: Detecting Strange Attractors in Turbulence. In Dynamical Systems and Turbulence, Warwick, 1980, ed. D.A. Rand, and L.S. Young, Lecture Notes in Mathematics, 898, Springer-Verlag, New York, 366–381, 1980.CrossRefGoogle Scholar
  25. [6.25]
    T. Sauer, J. Yorke and M. Casdagli: Embedology. J. Stat.Phys., 65, 579–616, 1991.MathSciNetMATHCrossRefGoogle Scholar
  26. [6.26]
    M. Hénon: A Two-Dimensional Mapping with a Strange Attractor. Comm. Math. Phys., 50, 69, 1976.MathSciNetMATHCrossRefGoogle Scholar
  27. [6.27]
    M. Mackey and L. Glass: Oscillation and chaos in physiological control systems. Science 197, 287, 1977.CrossRefGoogle Scholar
  28. [6.28]
    A.M. Fraser and H.L. Swinney: Independent Coordinates for Strange Attractors from Mutual Information. Phys. Rev. A., 33, 1134, 1986.MathSciNetMATHCrossRefGoogle Scholar
  29. [6.29]
    R. Abraham and J. Marsden: Theoretical Mechanics. The Benjamin-Cummings Publishing Company, Inc., London, 1978.Google Scholar
  30. [6.30]
    C.L. Siegel: Symplectic Geometry. Amer. Jour. Math., 65, 1–86, 1943.CrossRefGoogle Scholar
  31. [6.31]
    Feng Kang, Qin Meng-zhao: The Symplectic Methods for the Computation of Hamiltonian Equations. In: Zhu You-lan, Guo Ben-yu, eds., Numerical Methods for Partial Differential Equations. Lecture Notes in Mathematics., 1297, 1–35. Springer-Verlag, New York, 1985.Google Scholar
  32. [6.32]
    S. Miesbach, H.J. Pesch: Symplectic phase flow approximation for the numerical integration of canonical systems. Numer. Math., 61, 501–521, 1992.MathSciNetMATHCrossRefGoogle Scholar
  33. [6.33]
    K. Hornik, M. Stinchcombe, H. White: Multilayer Feedforward Neural Networks are Universal Approximators. Neural Networks, 2, 359–366, 1989.CrossRefGoogle Scholar
  34. [6.34]
    J. Stoer and R. Bulirsch: Introduction to Numerical Analysis. Springer-Verlag, New York, 1993.MATHGoogle Scholar
  35. [6.35]
    A. Papoulis: Probability, Random Variables, and Stochastic Processes. Third Edition, McGraw-Hill, New York, 1991.Google Scholar
  36. [6.36]
    J. Atick and A. Redlich: Towards a theory of early visual processing. Neural Computation, 2, 308–320, 1990.CrossRefGoogle Scholar
  37. [6.37]
    J. Atick: Could Information Theory Provide an Ecological Theory of Sensory Processing. Network, 3, 213–251, 1992.MATHCrossRefGoogle Scholar
  38. [6.38]
    D.J. Field: Relation Between the Statistics of Natural Images and the Response Properties of Cortical Cells. J. Opt. Soc. Am. A, 4, 2379–2394, 1987.CrossRefGoogle Scholar
  39. [6.39]
    R. De Valois, H. Morgan and D. Snodderly: Psycophysical Studies of Monkey Vision: III Spatial Luminance Contrast Sensitivity Test of Macaque Retina and Human Observers. Invest. Opthalmol. Vis. Sci., 14, 75–81, 1974.Google Scholar

Copyright information

© Springer-Verlag New York, Inc. 1996

Authors and Affiliations

  • Gustavo Deco
    • 1
  • Dragan Obradovic
    • 1
  1. 1.Corporate Research and DevelopmentSiemens AGMunichGermany

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