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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
J. Atick and A. Redlich: What Does the Retina Know about Natural Scenes. Neural Computation, 4, 196–210, 1992.
A. Redlich: Redundancy Reduction as a Strategy for Unsupervised Learning. Neural Computation, 5, 289–304, 1993.
A. Redlich: Supervised Factorial Learning. Neural Computation, 5, 750–766, 1993.
J. G. Taylor and S. Coombes: Learning Higher Order Correlations. Neural Networks, 6, 423–427,1993.
G. Deco and Brauer G: Nonlinear Higher-Order Statistical Decorrelation by Volume-Conserving Neural Architectures. Neural Networks, 8, 525–535, 1995.
G. Deco and B. Schürmann: Learning Time Series Evolution by Unsupervised Extraction of Correlations. Physical Review E, 51, 1780–1790, 1995.
G. Deco, L. Parra and S. Miesbach: Redundancy Reduction with Information-Preserving Nonlinear Maps. Network: Computation in Neural Systems, 6, 61–72, 1995.
J. Rubner and P. Tavan: A Self-Organization Network for Principal-Component Analysis. Europhysics Letters, 10, 693–698, 1989.
R. Durbin and D. Rumelhart: Product Units: A Computationally Powerful and Biologically Plausible Extension to Backpropagation Networks. Neural Computation, 1, 133–142, 1989.
C. Beck and F. Schlögl: Thermodynamics of Chaotic Systems. Cambridge Nonlinear Science Series, University Press, Cambridge, 1993.
P. Grassberger and I. Procaccia. Characterization of Strange Attractors. Phys. Rev. Lett., 50, 346, 1983.
J.P. Eckmann and D. Ruelle. Ergodic Theory of Chaos and Strange Attractors. Rev. Mod. Phys., 57, 617–656, 1985.
J.P. Crutchfield and McNamara: Equations of Motion from Data Series. Complex Systems, 1, 417–452, 1987.
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.
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.
J. Farmer and J. Sidorowich. Predicting Chaotic Time Series. Phys. Rev. Letters, 59, 845, 1987.
M. Casdagli: Nonlinear Prediction of Chaotic Time Series. Physica D, 35, 335–356, 1989.
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.
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.
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.
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.
W. Liebert, K. Pawelzik and H.G. Schuster: Optimal Embedding of Chaotic Attractors from Topological Considerations. Europhysics Lett., 14, 521–526, 1991.
K. Pawelzik and H.G. Schuster: Unstable Periodic Orbits and Prediction. Phys. Rev. A, 43, 1808–1812, 1991.
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.
T. Sauer, J. Yorke and M. Casdagli: Embedology. J. Stat.Phys., 65, 579–616, 1991.
M. Hénon: A Two-Dimensional Mapping with a Strange Attractor. Comm. Math. Phys., 50, 69, 1976.
M. Mackey and L. Glass: Oscillation and chaos in physiological control systems. Science 197, 287, 1977.
A.M. Fraser and H.L. Swinney: Independent Coordinates for Strange Attractors from Mutual Information. Phys. Rev. A., 33, 1134, 1986.
R. Abraham and J. Marsden: Theoretical Mechanics. The Benjamin-Cummings Publishing Company, Inc., London, 1978.
C.L. Siegel: Symplectic Geometry. Amer. Jour. Math., 65, 1–86, 1943.
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.
S. Miesbach, H.J. Pesch: Symplectic phase flow approximation for the numerical integration of canonical systems. Numer. Math., 61, 501–521, 1992.
K. Hornik, M. Stinchcombe, H. White: Multilayer Feedforward Neural Networks are Universal Approximators. Neural Networks, 2, 359–366, 1989.
J. Stoer and R. Bulirsch: Introduction to Numerical Analysis. Springer-Verlag, New York, 1993.
A. Papoulis: Probability, Random Variables, and Stochastic Processes. Third Edition, McGraw-Hill, New York, 1991.
J. Atick and A. Redlich: Towards a theory of early visual processing. Neural Computation, 2, 308–320, 1990.
J. Atick: Could Information Theory Provide an Ecological Theory of Sensory Processing. Network, 3, 213–251, 1992.
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.
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.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1996 Springer-Verlag New York, Inc.
About this chapter
Cite this chapter
Deco, G., Obradovic, D. (1996). Nonlinear Feature Extraction: Deterministic Neural Networks. In: An Information-Theoretic Approach to Neural Computing. Perspectives in Neural Computing. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-4016-7_6
Download citation
DOI: https://doi.org/10.1007/978-1-4612-4016-7_6
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4612-8469-7
Online ISBN: 978-1-4612-4016-7
eBook Packages: Springer Book Archive