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On Linguistic Variables and Sparse Representations

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Enric Trillas: A Passion for Fuzzy Sets

Part of the book series: Studies in Fuzziness and Soft Computing ((STUDFUZZ,volume 322))

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Abstract

Linguistic variables can be seen as dictionaries to represent data. In fields as Signal Processing or Machine Learning is usual to use or to search redundant dictionaries to promote sparse representations. This kind of representations present several interesting properties as a high generalization capacity, simplification and economy, among others. In this work, a revision of the main methods to obtain sparse representations and their possible application to model with linguistic variables and Fuzzy Rule Systems is done.

Author acknowledges the support of the Spanish Ministry for Economy and Innovation and the European Regional Development Fund (ERDF/FEDER) under grant TIN2011-29827-C02-02.

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Notes

  1. 1.

    This is the model we are consider here, however it is not a essential condition to have a linguistic variable in each dimension.

References

  1. Allen, J.B., Rabiner, L.: A unified approach to short-time Fourier analysis and synthesis. Proc. IEEE 65(11), 1558–1564 (1977)

    Article  Google Scholar 

  2. Bouchon-Meunier, B., Yia, Y.: Linguistic modifiers and imprecise categories. Int. J. Intell. Syst. 7, 25–36 (1992)

    Article  MATH  Google Scholar 

  3. Candès, E.J., Donoho, D.L.: Ridgelets: a key to higher-dimensional intermittency? Philos. Trans. R. Soc. Lond. Ser. A: Math. Phys. Eng. Sci. 357(1760), 2495–2509 (1999)

    Article  MATH  Google Scholar 

  4. Casillas, J., Cordón, O., Herrera, F., Magdalena, L. (eds.): Interpretability Issues in Fuzzy Modeling, Studies in Fuzziness and Soft Computing, vol. 128. Springer (2003)

    Google Scholar 

  5. Chandrasekaran, V., Wakin, M.B., Baron, D., Baraniuk, R.G.: Representation and compression of multidimensional piecewise functions using surflets. IEEE Trans. Inf. Theor. 55(1), 374–400 (2009)

    Article  MathSciNet  Google Scholar 

  6. Chen, S.S., Donoho, D.L., Saunders, M.A.: Atomic decomposition by basis pursuit. SIAM J. Sci. Comput. 20, 33–61 (1998)

    Article  MathSciNet  Google Scholar 

  7. Cooley, J.W.: An algorithm for the machine calculation of complex Fourier series. Math. Comput. 19(90), 297–301 (1965)

    Article  MATH  MathSciNet  Google Scholar 

  8. Cordón, O.: A historical review of evolutionary learning methods for Mamdani-type fuzzy rule-based systems: designing interpretable genetic fuzzy systems. Int. J. Approx. Reason. 52(6), 894–913 (2011)

    Article  Google Scholar 

  9. Daubechies, I.: Ten Lectures on Wavelets. CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM (1992)

    Google Scholar 

  10. Daubechies, I., Defrise, M., De Mol, C.: An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Commun. Pure Appl. Math. 57(11), 1413–1457 (2004)

    Article  MATH  Google Scholar 

  11. Davis, G., Mallat, S., Avellaneda, M.: Adaptive greedy approximations. Constr. Approx. 13(1), 57–98 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  12. Donoho, D.L., Huo, X.: Uncertainty principles and ideal atomic decomposition. IEEE Trans. Inf. Theor. 47(7), 2845–2862 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  13. Donoho, D.L., Elad, M.: Optimally sparse representation in general (nonorthogonal) dictionaries via \(\mathit{l}_1\) minimization. Proc. Natl. Acad. Sci. 100(5), 2197–2202 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  14. Donoho, D., Tsaig, Y., Drori, I., Starck, J.L.: Sparse solution of underdetermined systems of linear equations by stagewise orthogonal matching pursuit. IEEE Trans. Inf. Theor. 58(2), 1094–1121 (2012)

    Article  MathSciNet  Google Scholar 

  15. Efron, B., Hastie, T., Johnstone, I., Tibshirani, R.: Least angle regression (with discussion). Ann. Stat. 32(2), 407–451 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  16. Elad, M.: Sparse and Redundant Representations: From Theory to Applications in Signal and Image Processing, 1st edn. Springer Publishing Company, Incorporated (2010)

    Book  Google Scholar 

  17. Freund, Y., Schapire, R.E.: A decision-theoretic generalization of on-line learning and an application to boosting. J. Comput. Syst. Sci. 55(1), 119–139 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  18. Gacto, M., Alcalá, R., Herrera, F.: Interpretability of linguistic fuzzy rule-based systems: an overview of interpretability measures. Inf. Sci. 181(20), 4340–4360 (2011)

    Article  Google Scholar 

  19. Ho, N.C., Nam, H.V.: An algebraic approach to linguistic hedges in Zadeh’s fuzzy logic. Fuzzy Sets Syst. 129, 229–254 (2002)

    Google Scholar 

  20. Kar, S., Das, S., Ghosh, P.K.: Applications of neuro fuzzy systems: a brief review and future outline. Appl. Soft Comput. 15, 243–259 (2014)

    Article  Google Scholar 

  21. Kóczy, L., Hirota, K.: Interpolative reasoning with insufficient evidence in sparse fuzzy rule bases. Inf. Sci. 71(1–2), 169–201 (1993)

    Article  MATH  Google Scholar 

  22. Lakoff, G.: Hedges: A study of meaning criteria and the logic of fuzzy concepts. J. Philos. Log. 2, 458–508 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  23. Lee, H., Battle, A., Raina, R., Ng, A.: Efficient sparse coding algorithms. In: Advances in Neural Information Processing Systems, pp. 801–808 (2006)

    Google Scholar 

  24. Lughofer, E., Kindermann, S.: SparseFIS: data-driven learning of fuzzy systems with sparsity constraints. IEEE Trans. Fuzzy Syst. 18(2), 396–411 (2010)

    Google Scholar 

  25. Lughofer, E.: Extensions of vector quantization for incremental clustering. Pattern Recognit. 41(3), 995–1011 (2008)

    Article  MATH  Google Scholar 

  26. Luo, M., Sun, F., Liu, H.: Hierarchical sparse representation for T-S fuzzy systems identification. IEEE Trans. Fuzzy Syst. 21(6), 1032–1043 (2013)

    Article  Google Scholar 

  27. Mairal, J.: Sparse coding for machine learning, image processing and computer vision. Ph.D. thesis, Ecole Normale Supérieure de Cachan (2010)

    Google Scholar 

  28. Mallat, S.: A Wavelet Tour of Signal Processing. Academic Press (1999)

    Google Scholar 

  29. Mallat, S.G., Zhang, Z.: Matching pursuits with time-frequency dictionaries. IEEE Trans. Signal Proc. 41(12), 3397–3415 (1993)

    Article  MATH  Google Scholar 

  30. Lopez de Mantaras, R., Trillas, E.: Towards a measure of the degree of synonymy between concepts. In: Sánchez, E. (ed.) Fuzzy Information, Knowledge Representation and Decision Analysis. Pergamon Press, Inc. (1982)

    Google Scholar 

  31. Needell, D., Tropp, J.A.: CoSaMP: iterative signal recovery from incomplete and inaccurate samples. Appl. Comput. Harmon. Anal. 26(3), 301–321 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  32. Olshausen, B.A., Field, D.: Emergence of simple-cell receptive field properties by learning a sparse code for natural images. Nature 381(6583), 607–609 (1996)

    Article  Google Scholar 

  33. Ovchinnikov, S.: Representations of synonyms and antonyms by automorphisms in fuzzy set theory. Stochastica V(2), 95–107 (1981)

    Google Scholar 

  34. Pati, Y.C., Rezaiifar, R., Krishnaprasad, P.: Orthogonal matching pursuit: Recursive function approximation with applications to wavelet decomposition. In: 1993 Conference Record of The Twenty-Seventh Asilomar Conference on Signals, Systems and Computers, pp. 40–44. IEEE (1993)

    Google Scholar 

  35. de Soto, A.R.: A hierarchical model of a linguistic variable. Inf. Sci. 181(20), 4394–4408 (2011)

    Article  Google Scholar 

  36. de Soto, A.R., Trillas, E.: On antonym and negate in fuzzy logic. Int. J. Intell. Syst. 14, 295–303 (1999)

    Article  MATH  Google Scholar 

  37. Starck, J.L., Candès, E.J., Donoho, D.L.: The curvelet transform for image denoising. IEEE Trans. Image Process. 11(6), 670–684 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  38. Tibshirani, R.: Regression shrinkage and selection via the Lasso. J. R. Stat. Soc. Ser. B (Methodological) 267–288 (1996)

    Google Scholar 

  39. Trillas, E.: Sobre funciones de negación en la teoría de conjuntos difusos. Stochastica II I(1), 47–60 (1979), in Spanish

    Google Scholar 

  40. Trillas, E., Moraga, C., Guadarrama, S., Cubillo, S., Castiñeira, E.: Computing with antonyms. In: Nikravesh, M., Kacprzyk, J., Zadeh, L. (eds.) Forging New Frontiers: Fuzzy Pioneers I, Studies in Fuzziness and Soft Computing, vol. 217, pp. 133–153. Springer, Berlin / Heidelberg (2007)

    Chapter  Google Scholar 

  41. Trillas, E.: On negation functions in fuzzy set theory. In: Barro, S. et al. (eds.) Advances in Fuzzy Logic. Universidade de Santiago de Compostela, Spain (1998)

    Google Scholar 

  42. Trillas, E., Riera, T.: Towards a representation of synonyms and antonyms by fuzzy sets. Busefal 5, 42–68 (1981)

    Google Scholar 

  43. Tropp, J., Wright, S.: Computational methods for sparse solution of linear inverse problems. Proc. IEEE 98(6), 948–958 (2010)

    Article  Google Scholar 

  44. Tropp, J.A.: Greed is good: algorithmic results for sparse approximation. IEEE Trans. Inf. Theor. 50(10), 2231–2242 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  45. Xiang, Z.J., Ramadge, P.J.: Sparse boosting. In: IEEE International Conference on Acoustics, Speech, and Signal Processing, 1625–1628 (2009)

    Google Scholar 

  46. Xiang, Z.J., Xu, H., Ramadge, P.J.: Learning sparse representations of high dimensional data on large scale dictionaries. Advances in Neural Information Processing Systems (2011)

    Google Scholar 

  47. Zadeh, L.A.: A fuzzy-set-theoretic interpretation of linguistic hedges. J. Cybern. 2, 4–34 (1972)

    Article  MathSciNet  Google Scholar 

  48. Zadeh, L.: The concept of a linguistic variable and its application to approximate reasoning. part i. Inf. Sci. 8, 199–249 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  49. Zadeh, L.: The concept of a linguistic variable and its application to approximate reasoning. part I, II and III. In: Yager, R., Ovchinnikov, S., Tong, R., Nguyen, H. (eds.) Fuzzy Sets and Applications: Selected Papers by L.A. Zadeh, pp. 219–366. Wiley (1987)

    Google Scholar 

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Acknowledgments

Devoted to Professor Enric Trillas for his friendship and example.

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Authors and Affiliations

  1. Mechanical, Computer Science and Aerospace Engineering Department, University of León, Leon, Spain

    Adolfo R. de Soto

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  1. Adolfo R. de Soto
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Correspondence to Adolfo R. de Soto .

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Editors and Affiliations

  1. European Centre for Soft Computing, Asturias, Spain

    Luis Magdalena

  2. Depto. de Ciencias de la Computación e Inteligencia Artificial, University of Granada, Granada, Spain

    Jose Luis Verdegay

  3. Spanish Council for Scientific Research (CSIC), Artificial Intelligence Research Institute (IIIA-CSIC), Catalonia, Spain

    Francesc Esteva

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de Soto, A.R. (2015). On Linguistic Variables and Sparse Representations. In: Magdalena, L., Verdegay, J., Esteva, F. (eds) Enric Trillas: A Passion for Fuzzy Sets. Studies in Fuzziness and Soft Computing, vol 322. Springer, Cham. https://doi.org/10.1007/978-3-319-16235-5_14

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  • DOI: https://doi.org/10.1007/978-3-319-16235-5_14

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