Skip to main content
SpringerLink
Log in

Convergence in Orlicz spaces by means of the multivariate max-product neural network operators of the Kantorovich type and applications

  • Original Article
  • Published:
Neural Computing and Applications Aims and scope Submit manuscript

Abstract

In this paper, convergence results in a multivariate setting have been proved for a family of neural network operators of the max-product type. In particular, the coefficients expressed by Kantorovich type means allow to treat the theory in the general frame of the Orlicz spaces, which includes as particular case the \(L^p\)-spaces. Examples of sigmoidal activation functions are discussed, for the above operators in different cases of Orlicz spaces. Finally, concrete applications to real-world cases have been presented in both univariate and multivariate settings. In particular, the case of reconstruction and enhancement of biomedical (vascular) image has been discussed in detail.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  1. Adler A, Guardo R (1994) A neural network image reconstruction technique for electrical impedance tomography. IEEE Trans Med Imaging 13(4):594–600

    Article  Google Scholar 

  2. Angeloni L, Costarelli D, Vinti G (2018) A characterization of the convergence in variation for the generalized sampling series. Annales Academiae Scientiarum Fennicae Mathematica 43:755–767

    Article  MathSciNet  MATH  Google Scholar 

  3. Angeloni L, Vinti G (2009) Convergence and rate of approximation for linear integral operators in \(BV^\varphi \)-spaces in multidimensional setting. J Math Anal Appl 349:317–334

    MathSciNet  MATH  Google Scholar 

  4. Angeloni L, Vinti G (2010) Approximation with respect to Goffman–Serrin variation by means of non-convolution type integral operators. Numer Funct Anal Optim 31:519–548

    Article  MathSciNet  MATH  Google Scholar 

  5. Angeloni L, Vinti G (2014) Convergence and rate of approximation in BV\(\varphi ({\mathbb{R}}^N_+)\) for a class of Mellin integral operators. Atti Accad Naz Lincei Rend Lincei Mat Appl 25:217–232

    MathSciNet  MATH  Google Scholar 

  6. Asdrubali F, Baldinelli G, Bianchi F, Costarelli D, Rotili A, Seracini M, Vinti G (2018) Detection of thermal bridges from thermographic images by means of image processing approximation algorithms. Appl Math Comput 317:160–171

    MathSciNet  MATH  Google Scholar 

  7. Asdrubali F, Baldinelli G, Bianchi F, Costarelli D, Evangelisti L, Rotili A, Seracini M, Vinti G (2018) A model for the improvement of thermal bridges quantitative assessment by infrared thermography. Appl Energy 211:854–864

    Article  MATH  Google Scholar 

  8. Ball KR, Grant C, Mundy WR, Shafera TJ (2017) A multivariate extension of mutual information for growing neural networks. Neural Netw 95:29–43

    Article  Google Scholar 

  9. Bardaro C, Karsli H, Vinti G (2011) Nonlinear integral operators with homogeneous kernels: pointwise approximation theorems. Appl Anal 90(3–4):463–474

    Article  MathSciNet  MATH  Google Scholar 

  10. Bardaro C, Musielak J, Vinti G (2003) Nonlinear integral operators and applications. Series in nonlinear analysis and applications 9. W. De Gruyter & Co., Berlin

    Book  MATH  Google Scholar 

  11. Bartoccini B, Costarelli D, Vinti G (2018) Extension of saturation theorems for the sampling Kantorovich operators. Complex Anal Oper Theory. https://doi.org/10.1007/s11785-018-0852-z

    Article  MATH  Google Scholar 

  12. Bede B, Coroianu L, Gal SG (2016) Approximation by max-product type operators. Springer International Publishing, Berlin. https://doi.org/10.1007/978-3-319-34189-7

    Book  MATH  Google Scholar 

  13. Boccuto A, Candeloro D, Sambucini AR (2017) \(L^p\) spaces in vector lattices and applications. Math Slov 67(6):1409–1426. https://doi.org/10.1515/ms-2017-0060

    MATH  Google Scholar 

  14. Bono-Nuez A, Bernal-Ruíz C, Martín-del-Brío B, Pérez-Cebolla FJ, Martínez-Iturbe A (2017) Recipient size estimation for induction heating home appliances based on artificial neural networks. Neural Comput Appl 28(11):3197–3207

    Article  Google Scholar 

  15. Candeloro D, Sambucini AR (2015) Filter convergence and decompositions for vector lattice-valued measures. Mediterr J Math 12(3):621–637. https://doi.org/10.1007/s00009-014-0431-0

    Article  MathSciNet  MATH  Google Scholar 

  16. Cao F, Chen Z (2009) The approximation operators with sigmoidal functions. Comput Math Appl 58(4):758–765

    Article  MathSciNet  MATH  Google Scholar 

  17. Cao F, Chen Z (2012) The construction and approximation of a class of neural networks operators with ramp functions. J Comput Anal Appl 14(1):101–112

    MathSciNet  MATH  Google Scholar 

  18. Cao F, Liu B, Park DS (2013) Image classification based on effective extreme learning machine. Neurocomputing 102:90–97

    Article  Google Scholar 

  19. Cheang GHL (2010) Approximation with neural networks activated by ramp sigmoids. J Approx Theory 162:1450–1465

    Article  MathSciNet  MATH  Google Scholar 

  20. Coroianu L, Gal SG (2014) Saturation and inverse results for the Bernstein max-product operator. Period Math Hungar 69:126–133

    Article  MathSciNet  MATH  Google Scholar 

  21. Coroianu L, Gal SG (2017) \(L^p\)-approximation by truncated max-product sampling operators of Kantorovich-type based on Fejer kernel. J Integral Equ Appl 29(2):349–364

    MATH  Google Scholar 

  22. Costarelli D, Minotti AM, Vinti G (2017) Approximation of discontinuous signals by sampling Kantorovich series. J Math Anal Appl 450(2):1083–1103

    Article  MathSciNet  MATH  Google Scholar 

  23. Costarelli D, Sambucini AR (2018) Approximation results in Orlicz spaces for sequences of Kantorovich max-product neural network operators. Results Math 73(1):15. https://doi.org/10.1007/s00025-018-0799-4

    Article  MathSciNet  MATH  Google Scholar 

  24. Costarelli D, Spigler R (2015) How sharp is the Jensen inequality? J Inequal Appl 69:1–10

    MathSciNet  MATH  Google Scholar 

  25. Costarelli D, Spigler R (2018) Solving numerically nonlinear systems of balance laws by multivariate sigmoidal functions approximation. Comput Appl Math 37(1):99–133

    Article  MathSciNet  MATH  Google Scholar 

  26. Costarelli D, Vinti G (2016) Approximation by max-product neural network operators of Kantorovich type. Results Math 69(3):505–519

    Article  MathSciNet  MATH  Google Scholar 

  27. Costarelli D, Vinti G (2016) Max-product neural network and quasi-interpolation operators activated by sigmoidal functions. J Approx Theory 209:1–22

    Article  MathSciNet  MATH  Google Scholar 

  28. Costarelli D, Vinti G (2016) Pointwise and uniform approximation by multivariate neural network operators of the max-product type. Neural Netw 81:81–90

    Article  MATH  Google Scholar 

  29. Costarelli D, Vinti G (2017) Saturation classes for max-product neural network operators activated by sigmoidal functions. Results Math 72(3):1555–1569

    Article  MathSciNet  MATH  Google Scholar 

  30. Costarelli D, Vinti G (2017) Convergence for a family of neural network operators in Orlicz spaces. Mathematische Nachrichten 290(2–3):226–235

    Article  MathSciNet  MATH  Google Scholar 

  31. Costarelli D, Vinti G (2017) Convergence results for a family of Kantorovich max-product neural network operators in a multivariate setting. Mathematica Slovaca 67(6):1469–1480

    Article  MathSciNet  MATH  Google Scholar 

  32. Costarelli D, Vinti G (2018) Estimates for the neural network operators of the max-product type with continuous and p-integrable functions. Results Math 73(1):12. https://doi.org/10.1007/s00025-018-0790-0

    Article  MathSciNet  MATH  Google Scholar 

  33. Costarelli D, Vinti G (2018) An inverse result of approximation by sampling Kantorovich series. Proc Edinb Math Soc. https://doi.org/10.1017/S0013091518000342

    Article  MATH  Google Scholar 

  34. Egmont-Petersena M, de Ridderb D, Handels H (2002) Image processing with neural networks—a review. Pattern Recognit 35:2279–2301

    Article  MATH  Google Scholar 

  35. Gnecco G (2012) A comparison between fixed-basis and variable-basis schemes for function approximation and functional optimization. J Appl Math. https://doi.org/10.1155/2012/806945

    Article  MathSciNet  MATH  Google Scholar 

  36. Gnecco G, Sanguineti M (2011) On a variational norm tailored to variable-basis approximation schemes. IEEE Trans Inf Theory 57:549–558

    Article  MathSciNet  MATH  Google Scholar 

  37. Goh ATC (1995) Back-propagation neural networks for modeling complex systems. Artif Intell Eng 9:143–151

    Article  Google Scholar 

  38. Gotleyb D, Lo Sciuto G, Napoli C, Shikler R, Tramontana E, Wozniak M (2016) Characterization and modeling of organic solar cells by using radial basis neural networks. Artif Intell Soft Comput. https://doi.org/10.1007/978-3-319-39378-0_9

    Article  Google Scholar 

  39. Guliyev NJ, Ismailov VE (2018) On the approximation by single hidden layer feedforward neural networks with fixed weights. Neural Networks 98:296–304

    Article  Google Scholar 

  40. Guliyev NJ, Ismailov VE (2018) Approximation capability of two hidden layer feedforward neural networks with fixed weights. Neurocomputing 316:262–269

    Article  Google Scholar 

  41. Iliev A, Kyurkchiev N, Markov S (2015) On the approximation of the cut and step functions by logistic and Gompertz functions. Biomath. https://doi.org/10.11145/j.biomath.2015.10.101

    Article  MathSciNet  MATH  Google Scholar 

  42. Kainen PC, Kurkova V, Sanguineti M (2009) Complexity of Gaussian-radial-basis networks approximating smooth functions. J Complex 25(1):63–74

    Article  MathSciNet  MATH  Google Scholar 

  43. Lai G, Liu Z, Zhang Y, Philip Chen CL (2016) Adaptive position/attitude tracking control of aerial robot with unknown inertial matrix based on a new robust neural identifier. IEEE Trans Neural Netw Learn Syst 27(1):18–31

    Article  MathSciNet  Google Scholar 

  44. Liu P, Wang J, Zeng Z (2017) Multistability of delayed recurrent neural networks with Mexican hat activation functions. Neural Comput 29(2):423–457

    Article  MathSciNet  MATH  Google Scholar 

  45. Livingstone DJ (2008) Artificial neural networks: methods and applications (methods in molecular biology). Humana Press, New York

    Google Scholar 

  46. Maiorov V (2006) Approximation by neural networks and learning theory. J Complex 22(1):102–117

    Article  MathSciNet  MATH  Google Scholar 

  47. Musielak J (1983) Orlicz spaces and modular spaces. Lecture notes in mathematics, vol 1034. Springer, Berlin

    Book  MATH  Google Scholar 

  48. Musielak J, Orlicz W (1959) On modular spaces. Studia Math 28:49–65

    Article  MathSciNet  MATH  Google Scholar 

  49. Olivera JJ (2017) Global exponential stability of nonautonomous neural network models with unbounded delays. Neural Netw 96:71–79

    Article  Google Scholar 

  50. Rister B, Rubin DL (2017) Piecewise convexity of artificial neural networks. Neural Netw 94:34–45

    Article  Google Scholar 

  51. Sahoo A, Xu H, Jagannathan S (2016) Adaptive neural network-based event-triggered control of single-input single-output nonlinear discrete-time systems. IEEE Trans Neural Netw Learn Syst 27(1):151–164

    Article  MathSciNet  Google Scholar 

  52. Stamov G, Stamova I (2017) Impulsive fractional-order neural networks with time-varying delays: almost periodic solutions. Neural Comput Appl 28(11):3307–3316

    Article  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the referees for their useful suggestions which led us to insert the section devoted to real-world applications.

Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Sciences, Università degli Studi di Perugia, 06123, Perugia, Italy

    Danilo Costarelli, Anna Rita Sambucini & Gianluca Vinti

Authors
  1. Danilo Costarelli
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Anna Rita Sambucini
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Gianluca Vinti
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Danilo Costarelli.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Ethical statement

Ethical approval was waived considering that the CT images analyzed were anonymized and the results did not influence any clinical judgment.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). Moreover, the first and the second authors of the paper have been partially supported within the 2018 GNAMPA-INdAM Project “Dinamiche non autonome, analisi reale e applicazioni,” while the second and the third authors within the project: Ricerca di Base 2017 dell’Università degli Studi di Perugia—“Metodi di teoria degli operatori e di Analisi Reale per problemi di approssimazione ed applicazioni.”

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Costarelli, D., Sambucini, A.R. & Vinti, G. Convergence in Orlicz spaces by means of the multivariate max-product neural network operators of the Kantorovich type and applications. Neural Comput & Applic 31, 5069–5078 (2019). https://doi.org/10.1007/s00521-018-03998-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00521-018-03998-6

Keywords

Mathematics Subject Classification

Search

Navigation