Foundations of Computational Mathematics

, Volume 10, Issue 1, pp 67–91 | Cite as

Mathematics of the Neural Response

  • S. Smale
  • L. Rosasco
  • J. Bouvrie
  • A. Caponnetto
  • T. Poggio
Article

Abstract

We propose a natural image representation, the neural response, motivated by the neuroscience of the visual cortex. The inner product defined by the neural response leads to a similarity measure between functions which we call the derived kernel. Based on a hierarchical architecture, we give a recursive definition of the neural response and associated derived kernel. The derived kernel can be used in a variety of application domains such as classification of images, strings of text and genomics data.

Keywords

Unsupervised learning Computer vision Kernels 

Mathematics Subject Classification (2000)

68Q32 68T45 68T10 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    N. Aronszajn, Theory of reproducing kernels, Trans. Am. Math. Soc. 68, 337–404 (1950). MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    A. Caponnetto, T. Poggio, S. Smale, On a model of visual cortex: learning invariance and selectivity from image sequences. CBCL paper 272 / CSAIL technical report 2008-030, MIT, Cambridge, MA (2008). Google Scholar
  3. 3.
    T.M. Cover, J.A. Thomas, Elements of Information Theory (Wiley, New York, 1991). MATHCrossRefGoogle Scholar
  4. 4.
    N. Cristianini, J. Shawe-Taylor, An Introduction to Support Vector Machines (Cambridge University Press, Cambridge, 2000). Google Scholar
  5. 5.
    F. Cucker, D.X. Zhou, Learning Theory: An Approximation Theory Viewpoint. Cambridge Monographs on Applied and Computational Mathematics (Cambridge University Press, Cambridge, 2007). MATHGoogle Scholar
  6. 6.
    D.A. Forsyth, J. Ponce, Computer Vision: A Modern Approach (Prentice Hall, New York, 2002). Google Scholar
  7. 7.
    K. Fukushima, Neocognitron: A self-organizing neural network model for a mechanism of pattern recognition unaffected by shift in position, Biol. Cybern. 36, 193–202 (1980). MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Y. LeCun, L. Bottou, Y. Bengio, P. Haffner, Gradient-based learning applied to document recognition, Proc. IEEE 86(11), 2278–2324 (1998). CrossRefGoogle Scholar
  9. 9.
    T. Poggio, S. Smale, The mathematics of learning: Dealing with data, Not. Am. Math. Soc. (AMS), 50(5) (2003). Google Scholar
  10. 10.
    M. Riesenhuber, T. Poggio, Hierarchical models of object recognition in cortex, Nat. Neurosci. 2, 1019–1025 (1999). CrossRefGoogle Scholar
  11. 11.
    B. Schölkopf, A.J. Smola, Learning with Kernels (MIT Press, Cambridge, 2002). Google Scholar
  12. 12.
    T. Serre, M. Kouh, C. Cadieu, U. Knoblich, G. Kreiman, T. Poggio, A theory of object recognition: computations and circuits in the feedforward path of the ventral stream in primate visual cortex. AI Memo 2005-036 / CBCL Memo 259, MIT, Cambridge, MA (2005). Google Scholar
  13. 13.
    T. Serre, M. Kouh, C. Cadieu, U. Knoblich, G. Kreiman, T. Poggio, A quantitative theory of immediate visual recognition, Prog. Brain Res. 165, 33–56 (2007). CrossRefGoogle Scholar
  14. 14.
    T. Serre, A. Oliva, T. Poggio, A feedforward architecture accounts for rapid categorization, Proc. Natl. Acad. Sci. 104, 6424–6429 (2007). CrossRefGoogle Scholar
  15. 15.
    T. Serre, L. Wolf, S. Bileschi, M. Riesenhuber, T. Poggio, Robust object recognition with cortex-like mechanisms, IEEE Trans. Pattern Anal. Mach. Intell. 29, 411–426 (2007). CrossRefGoogle Scholar
  16. 16.
    J. Shawe-Taylor, N. Cristianini, Kernel Methods for Pattern Analysis (Cambridge University Press, Cambridge, 2004). Google Scholar
  17. 17.
    S. Smale, T. Poggio, A. Caponnetto, J. Bouvrie, Derived distance: towards a mathematical theory of visual cortex. CBCL paper, MIT, Cambridge, MA (November 2007). Google Scholar
  18. 18.
    S. Ullman, M. Vidal-Naquet, E. Sali, Visual features of intermediate complexity and their use in classification, Nat. Neurosci. 5(7), 682–687 (2002). Google Scholar
  19. 19.
    V.N. Vapnik, Statistical Learning Theory (Wiley, New York, 1998). MATHGoogle Scholar
  20. 20.
    H. Wersing, E. Koerner, Learning optimized features for hierarchical models of invariant recognition, Neural Comput. 15(7), 1559–1588 (2003). MATHCrossRefGoogle Scholar

Copyright information

© SFoCM 2009

Authors and Affiliations

  • S. Smale
    • 1
  • L. Rosasco
    • 2
  • J. Bouvrie
    • 3
  • A. Caponnetto
    • 4
  • T. Poggio
    • 5
  1. 1.Toyota Technological Institute at Chicago and University of CaliforniaBerkeleyUSA
  2. 2.CBCL, McGovern Institute, MIT & DISIUniversità di GenovaCambridgeUSA
  3. 3.CBCL, Brain and Cognitive SciencesMassachusetts Institute of TechnologyCambridgeUSA
  4. 4.Department of MathematicsCity University of Hong KongHong KongChina
  5. 5.CBCL, McGovern Institute, CSAIL, BCSMassachusetts Institute of TechnologyCambridgeUSA

Personalised recommendations