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Morphological Perceptrons: Geometry and Training Algorithms

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Mathematical Morphology and Its Applications to Signal and Image Processing (ISMM 2017)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 10225))

Abstract

Neural networks have traditionally relied on mostly linear models, such as the multiply-accumulate architecture of a linear perceptron that remains the dominant paradigm of neuronal computation. However, from a biological standpoint, neuron activity may as well involve inherently nonlinear and competitive operations. Mathematical morphology and minimax algebra provide the necessary background in the study of neural networks made up from these kinds of nonlinear units. This paper deals with such a model, called the morphological perceptron. We study some of its geometrical properties and introduce a training algorithm for binary classification. We point out the relationship between morphological classifiers and the recent field of tropical geometry, which enables us to obtain a precise bound on the number of linear regions of the maxout unit, a popular choice for deep neural networks introduced recently. Finally, we present some relevant numerical results.

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Notes

  1. 1.

    The term “tropical” was playfully introduced by French mathematicians in honor of the Brazilian theoretical computer scientist, Imre Simon. Another example of a tropical semiring is the \((\max , \times )\) semiring, also referred to as the subtropical semiring.

  2. 2.

    The matrix \(-{\varvec{A}}^T\), often denoted by \({\varvec{A}}^{\sharp }\) in the tropical geometry community, is sometimes called the Cuninghame-Green inverse of \({\varvec{A}}\).

References

  1. Allamigeon, X., Benchimol, P., Gaubert, S., Joswig, M.: Tropicalizing the simplex algorithm. SIAM J. Discret. Math. 29(2), 751–795 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  2. Araújo, R.D.A., Oliveira, A.L., Meira, S.R: A hybrid neuron with gradient-based learning for binary classification problems. In: Encontro Nacional de Inteligência Artificial-ENIA (2012)

    Google Scholar 

  3. Barrera, J., Dougherty, E.R., Tomita, N.S.: Automatic programming of binary morphological machines by design of statistically optimal operators in the context of computational learning theory. J. Electron. Imaging 6(1), 54–67 (1997)

    Article  Google Scholar 

  4. Butkovič, P.: Max-linear Systems: Theory and Algorithms. Springer Science & Business Media, Heidelberg (2010)

    Book  MATH  Google Scholar 

  5. Cuninghame-Green, R.A.: Minimax Algebra. Lecture Notes in Economics and Mathematical Systems, vol. 166. Springer, Heidelberg (1979)

    MATH  Google Scholar 

  6. Davidson, J.L., Hummer, F.: Morphology neural networks: an introduction with applications. Circ. Syst. Sig. Process. 12(2), 177–210 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  7. Diamond, S., Boyd, S.: CVXPY: a Python-embedded modeling language for convex optimization. J. Mach. Learn. Res. 17(83), 1–5 (2016)

    MathSciNet  MATH  Google Scholar 

  8. Gärtner, B., Jaggi, M.: Tropical support vector machines. Technical report ACS-TR-362502-01 (2008)

    Google Scholar 

  9. Gaubert, S., Katz, R.D.: Minimal half-spaces and external representation of tropical polyhedra. J. Algebraic Comb. 33(3), 325–348 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. Gondran, M., Minoux, M.: Graphs, Dioids and Semirings: New Models and Algorithms, vol. 41. Springer Science & Business Media, Heidelberg (2008)

    MATH  Google Scholar 

  11. Goodfellow, I.J., Warde-Farley, D., Mirza, M., Courville, A.C., Bengio, Y.: Maxout networks. ICML 3(28), 1319–1327 (2013)

    Google Scholar 

  12. LeCun, Y., Cortes, C., Burges, C.J.: The MNIST database of handwritten digits (1998)

    Google Scholar 

  13. Maclagan, D., Sturmfels, B.: Introduction to Tropical Geometry, vol. 161. American Mathematical Society, Providence (2015)

    MATH  Google Scholar 

  14. Maragos, P.: Morphological filtering for image enhancement and feature detection. In: Bovik, A.C. (ed.) The Image and Video Processing Handbook, 2nd edn, pp. 135–156. Elsevier Academic Press, Amsterdam (2005)

    Chapter  Google Scholar 

  15. Maragos, P.: Dynamical systems on weighted lattices: general theory. arXiv preprint arXiv:1606.07347 (2016)

  16. Montufar, G.F., Pascanu, R., Cho, K., Bengio, Y.: On the number of linear regions of deep neural networks. In: Advances in Neural Information Processing Systems, pp. 2924–2932 (2014)

    Google Scholar 

  17. Nair, V., Hinton, G.E.: Rectified linear units improve restricted Boltzmann machines. In: ICML 2010, pp. 807–814 (2010)

    Google Scholar 

  18. Ripley, B.D.: Pattern Recognition and Neural Networks. Cambridge University Press, Cambridge (2007)

    MATH  Google Scholar 

  19. Ritter, G.X., Sussner, P.: An introduction to morphological neural networks. In: 1996 Proceedings of the 13th International Conference on Pattern Recognition, vol. 4, pp. 709–717. IEEE (1996)

    Google Scholar 

  20. Ritter, G.X., Urcid, G.: Lattice algebra approach to single-neuron computation. IEEE Trans. Neural Netw. 14(2), 282–295 (2003)

    Article  Google Scholar 

  21. Rosenblatt, F.: The perceptron: a probabilistic model for information storage and organization in the brain. Psychol. Rev. 65(6), 386 (1958)

    Article  Google Scholar 

  22. Serra, J.: Image Analysis and Mathematical Morphology, vol. 1. Academic Press, Cambridge (1982)

    MATH  Google Scholar 

  23. Shen, X., Diamond, S., Gu, Y., Boyd, S.: Disciplined convex-concave programming. arXiv preprint arXiv:1604.02639 (2016)

  24. Sussner, P., Esmi, E.L.: Morphological perceptrons with competitive learning: lattice-theoretical framework and constructive learning algorithm. Inf. Sci. 181(10), 1929–1950 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  25. Sussner, P., Valle, M.E.: Gray-scale morphological associative memories. IEEE Trans. Neural Netw. 17(3), 559–570 (2006)

    Article  Google Scholar 

  26. Vanderbei, R.J., et al.: Linear Programming. Springer, Heidelberg (2015)

    Book  Google Scholar 

  27. Wolberg, W.H., Mangasarian, O.L.: Multisurface method of pattern separation for medical diagnosis applied to breast cytology. Proc. Nat. Acad. Sci. 87(23), 9193–9196 (1990)

    Article  MATH  Google Scholar 

  28. Xu, L., Crammer, K., Schuurmans, D.: Robust support vector machine training via convex outlier ablation. In: AAAI, vol. 6, pp. 536–542 (2006)

    Google Scholar 

  29. Yang, P.F., Maragos, P.: Min-max classifiers: learnability, design and application. Pattern Recogn. 28(6), 879–899 (1995)

    Article  Google Scholar 

  30. Yuille, A.L., Rangarajan, A.: The concave-convex procedure. Neural Comput. 15(4), 915–936 (2003)

    Article  MATH  Google Scholar 

  31. Ziegler, G.M.: Lectures on Polytopes, vol. 152. Springer Science & Business Media, Heidelberg (1995)

    MATH  Google Scholar 

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Acknowledgements

This work was partially supported by the European Union under the projects BabyRobot with grant H2020-687831 and I-SUPPORT with grant H2020-643666.

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Authors and Affiliations

  1. School of ECE, National Technical University of Athens, 15773, Athens, Greece

    Vasileios Charisopoulos & Petros Maragos

Authors
  1. Vasileios Charisopoulos
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  2. Petros Maragos
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Correspondence to Vasileios Charisopoulos .

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Editors and Affiliations

  1. MINES ParisTech, PSL-Research University, Fontainebleau, France

    Jesús Angulo

  2. MINES ParisTech, PSL-Research University , Fontainebleau, France

    Santiago Velasco-Forero

  3. MINES ParisTech, PSL-Research University, Fontainebleau, France

    Fernand Meyer

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Charisopoulos, V., Maragos, P. (2017). Morphological Perceptrons: Geometry and Training Algorithms. In: Angulo, J., Velasco-Forero, S., Meyer, F. (eds) Mathematical Morphology and Its Applications to Signal and Image Processing. ISMM 2017. Lecture Notes in Computer Science(), vol 10225. Springer, Cham. https://doi.org/10.1007/978-3-319-57240-6_1

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  • DOI: https://doi.org/10.1007/978-3-319-57240-6_1

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-57239-0

  • Online ISBN: 978-3-319-57240-6

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