Approximation by multivariate max-product Kantorovich-type operators and learning rates of least-squares regularized regression

Lucian Coroianu; Danilo Costarelli; Sorin G. Gal; Gianluca Vinti

doi:10.3934/cpaa.2020189

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August 2020, 19(8): 4213-4225. doi: 10.3934/cpaa.2020189

Approximation by multivariate max-product Kantorovich-type operators and learning rates of least-squares regularized regression

Lucian Coroianu ^1, , Danilo Costarelli ^2, , Sorin G. Gal ^1,, and Gianluca Vinti ^2,

1.	Department of Mathematics and Computer Science, University of Oradea, Oradea, Romania
2.	Department of Mathematics and Computer Science, University of Perugia, Perugia, Italy

^* Corresponding author

Received October 2019 Revised March 2020 Published May 2020

In a recent paper, for univariate max-product sampling operators based on general kernels with bounded generalized absolute moments, we have obtained several $ L^{p}_{\mu} $ convergence properties on bounded intervals or on the whole real axis. In this paper, firstly we obtain quantitative estimates with respect to a $ K $-functional, for the multivariate Kantorovich variant of these max-product sampling operators with the integrals written in terms of Borel probability measures. Applications of these approximation results to learning theory are obtained.

Keywords: Multivariate max-product sampling Kantorovich operators, Borel probability measures, multivariate generalized kernels, L_µ^p-norm, $ 1\le p <\infty $, $ K $-functional, learning theory, regularizing function, sample error, regularized error.

Mathematics Subject Classification: Primary: 41A35, 41A25, 41A63; Secondary: 62J02, 68T05.

Citation: Lucian Coroianu, Danilo Costarelli, Sorin G. Gal, Gianluca Vinti. Approximation by multivariate max-product Kantorovich-type operators and learning rates of least-squares regularized regression. Communications on Pure and Applied Analysis, 2020, 19 (8) : 4213-4225. doi: 10.3934/cpaa.2020189

References:

[1]	L. Angeloni, D. Costarelli and G. Vinti, A characterization of the convergence in variation for the generalized sampling series, Ann. Acad. Sci. Fenn. Math., 43 (2018), 755-767. doi: 10.5186/aasfm.2018.4343.
[2]	F. Asdrubali, G. Baldinelli, F. Bianchi, D. Costarelli, A. Rotili, M. Seracini and G. Vinti, Detection of thermal bridges from thermographic images by means of image processing approximation algorithms, Appl. Math. Comput., 317 (2018), 160-171. doi: 10.1016/j.amc.2017.08.058.
[3]	C. Bardaro, P. L. Butzer, R. L. Stens and G. Vinti, Kantorovich-type generalized sampling series in the setting of Orlicz spaces, Sampl. Theory Signal Image Process., 6 (2007), 19-52.
[4]	C. Bardaro and I. Mantellini, Generalized sampling approximation of bivariate signals: rate of pointwise convergence, Numer. Funct. Anal. Optim., 31 (2010), 131-154. doi: 10.1080/01630561003644702.
[5]	B. Bede, L. Coroianu and S. G. Gal, Approximation by Max-Product Type Operators, Springer, New York, 2016. doi: 10.1007/978-3-319-34189-7.
[6]	B. Bede, L. Coroianu and S. G. Gal, Approximation and shape preserving properties of the Bernstein operator of max-product kind, Int. J. Math. Math. Sci., 2009 (2009), Art. 590589, 26 pp. doi: 10.1155/2009/590589.
[7]	P. L. Butzer, A survey of the Whittaker-Shannon sampling theorem and some of its extensions, J. Math. Res. Expos., 3 (1983), 185-212.
[8]	P. L. Butzer, H. G. Feichtinger and K. Grochenig, Error analysis in regular and irregular sampling theory, Appl. Anal., 50 (1993), 167-189. doi: 10.1080/00036819308840192.
[9]	P. L. Butzer and R. J. Nessel, Fourier Analysis and Approximations, I, Academic Press, New York-London, 1971.
[10]	P. L. Butzer, S. Riesz and R. L. Stens, Approximation of continuous and discontinuous functions by generalized sampling series, J. Approx. Theory, 50 (1987), 25-39. doi: 10.1016/0021-9045(87)90063-3.
[11]	L. Coroianu, D. Costarelli, S. G. Gal and G. Vinti, The max-product generalized sampling operators: convergence and quantitative estimates, Appl. Math. Comput., 355 (2019), 173-183. doi: 10.1016/j.amc.2019.02.076.
[12]	L. Coroianu, D. Costarelli, S. G. Gal and G. Vinti, Approximation by max-product sampling Kantorovich operators with generalized kernels, Anal. Appl., 2019, in press. doi: 10.1142/S0219530519500155.
[13]	L. Coroianu and S. G. Gal, Approximation by nonlinear generalized sampling operators of max-product kind, Sampl. Theory Signal Image Process., 9 (2010), 59-75.
[14]	L. Coroianu and S. G. Gal, Approximation by max-product sampling operators based on sinc-type kernels, Sampl. Theory Signal Image Process., 10 (2011), 211-230.
[15]	L. Coroianu and S. G. Gal, Classes of functions with improved estimates in approximation by the max-product Bernstein operator, Anal. Appl., 9 (2011), 249-274. doi: 10.1142/S0219530511001856.
[16]	L. Coroianu and S. G. Gal, Approximation by truncated max-product operators of Kantorovich-type based on generalized $(\varphi, \psi)$-kernels, Math. Meth. Appl. Sci., 41 (2018), 7971-7984. doi: 10.1002/mma.5262.
[17]	L. Coroianu and S. G. Gal, $L^{p}$-approximation by truncated max-product sampling operators of Kantorovich-type based on Fejér kernel, J. Integr. Equ. Appl., 29 (2017), 349-364. doi: 10.1216/JIE-2017-29-2-349.
[18]	L. Coroianu and S. G. Gal, Saturation results for the truncated max-product sampling operators based on sinc and Fejér-type kernels, Sampl. Theory Signal Image Process., 11 (2012), 113-132.
[19]	D. Costarelli, A. M. Minotti and G. Vinti, Approximation of discontinuous signals by sampling Kantorovich series, J. Math. Anal. Appl., 450 (2017), 1083-1103. doi: 10.1016/j.jmaa.2017.01.066.
[20]	D. Costarelli and A. R. Sambucini, Approximation results in Orlicz spaces for sequences of Kantorovich max-product neural network operators, Results Math., 73 (2018), Art. 15. doi: 10.1007/s00025-018-0799-4.
[21]	D. Costarelli, A. R. Sambucini and G. Vinti, Convergence in Orlicz spaces by means of the multivariate max-product neural network operators of the Kantorovich type and applications, Neural Comput. Appl., 31 (2019), 5069-5078. doi: 10.1007/s00521-018-03998-6.
[22]	D. Costarelli and G. Vinti, Order of approximation for sampling Kantorovich type operators, J. Integr. Equ. Appl., 26 (2014), 345-368. doi: 10.1216/JIE-2014-26-3-345.
[23]	D. Costarelli and G. Vinti, Rate of approximation for multivariate sampling Kantorovich operators on some functions spaces, J. Integr. Equ. Appl., 26 (2014), 455-481. doi: 10.1216/JIE-2014-26-4-455.
[24]	D. Costarelli and G. Vinti, Approximation by max-product neural network operators of Kantorovich type, Results Math., 69 (2016), 505-519. doi: 10.1007/s00025-016-0546-7.
[25]	D. Costarelli and G. Vinti, Max-product neural network and quasi-interpolation operators activated by sigmoidal functions, J. Approx. Theory, 209 (2016), 1-22. doi: 10.1016/j.jat.2016.05.001.
[26]	D. Costarelli and G. Vinti, Pointwise and uniform approximation by multivariate neural network operators of the max-product type, Neural Netw., 81 (2016), 81-90.
[27]	D. Costarelli and G. Vinti, Convergence results for a family of Kantorovich max-product neural network operators in a multivariate setting, Math. Slovaca, 67 (2017), 1469-1480. doi: 10.1515/ms-2017-0063.
[28]	D. Costarelli and G. Vinti, Convergence for a family of neural network operators in Orlicz spaces, Math. Nachr., 290 (2017), 226-235. doi: 10.1002/mana.201600006.
[29]	D. Costarelli and G. Vinti, Estimates for the neural network operators of the max-product type with continuous and $p$-integrable functions, Results Math., 73 (2018), Art. 12. doi: 10.1007/s00025-018-0790-0.
[30]	D. Costarelli and G. Vinti, An inverse result of approximation by sampling Kantorovich series, Proc. Edinb. Math. Soc., 62 (2019), 265-280. doi: 10.1017/s0013091518000342.
[31]	S. Y. Güngör and N. Ispir, Approximation by Bernstein-Chlodowsky operators of max-product kind, Math. Commun., 23 (2018), 205-225.
[32]	A. Holhos, Weighted Approximation of functions by Meyer-König and Zeller operators of max-product type, Numer. Funct. Anal. Optim., 39 (2018), 689-703. doi: 10.1080/01630563.2017.1413386.
[33]	A. Holhos, Weighted approximation of functions by Favard operators of max-product type, Period. Math. Hungar., 77 (2018), 340-346. doi: 10.1007/s10998-018-0249-9.
[34]	B. Z. Li, Approximation by multivariate Bernstein-Durrmeyer operators and learning rates of least-squares regularized regression with multivariate polynomial kernels, J. Approx. Theory, 173 (2013), 33-55. doi: 10.1016/j.jat.2013.04.007.
[35]	O. Orlova and G. Tamberg, On approximation properties of generalized Kantorovich-type sampling operators, J. Approx. Theory, 201 (2016), 73-86. doi: 10.1016/j.jat.2015.10.001.
[36]	R. J. Ravier and R. S. Stichartz, Sampling theory with average values on the Sierpinski gasket, Constr. Approx., 44 (2016), 159-194. doi: 10.1007/s00365-016-9341-7.
[37]	R. L. Stens, Error estimates for sampling sums based on convolution integrals, Inform. Control, 45 (1980), 37-47. doi: 10.1016/S0019-9958(80)90857-8.
[38]	D. X. Zhou, Deep distributed convolutional neural networks: universality, Anal. Appl., 16 (2018), 895-919. doi: 10.1142/S0219530518500124.
[39]	D. X. Zhou, Universality of deep convolutional neural networks, Appl. Comput. Harmon. Anal., 48 (2020), 787-794. doi: 10.1016/j.acha.2019.06.004.
[40]	D. X. Zhou and K. Jetter, Approximation with polynomial kernels and SVM classifiers, Adv. Comput. Math., 25 (2006), 323-344. doi: 10.1007/s10444-004-7206-2.

show all references

References:

[1]	L. Angeloni, D. Costarelli and G. Vinti, A characterization of the convergence in variation for the generalized sampling series, Ann. Acad. Sci. Fenn. Math., 43 (2018), 755-767. doi: 10.5186/aasfm.2018.4343.
[2]	F. Asdrubali, G. Baldinelli, F. Bianchi, D. Costarelli, A. Rotili, M. Seracini and G. Vinti, Detection of thermal bridges from thermographic images by means of image processing approximation algorithms, Appl. Math. Comput., 317 (2018), 160-171. doi: 10.1016/j.amc.2017.08.058.
[3]	C. Bardaro, P. L. Butzer, R. L. Stens and G. Vinti, Kantorovich-type generalized sampling series in the setting of Orlicz spaces, Sampl. Theory Signal Image Process., 6 (2007), 19-52.
[4]	C. Bardaro and I. Mantellini, Generalized sampling approximation of bivariate signals: rate of pointwise convergence, Numer. Funct. Anal. Optim., 31 (2010), 131-154. doi: 10.1080/01630561003644702.
[5]	B. Bede, L. Coroianu and S. G. Gal, Approximation by Max-Product Type Operators, Springer, New York, 2016. doi: 10.1007/978-3-319-34189-7.
[6]	B. Bede, L. Coroianu and S. G. Gal, Approximation and shape preserving properties of the Bernstein operator of max-product kind, Int. J. Math. Math. Sci., 2009 (2009), Art. 590589, 26 pp. doi: 10.1155/2009/590589.
[7]	P. L. Butzer, A survey of the Whittaker-Shannon sampling theorem and some of its extensions, J. Math. Res. Expos., 3 (1983), 185-212.
[8]	P. L. Butzer, H. G. Feichtinger and K. Grochenig, Error analysis in regular and irregular sampling theory, Appl. Anal., 50 (1993), 167-189. doi: 10.1080/00036819308840192.
[9]	P. L. Butzer and R. J. Nessel, Fourier Analysis and Approximations, I, Academic Press, New York-London, 1971.
[10]	P. L. Butzer, S. Riesz and R. L. Stens, Approximation of continuous and discontinuous functions by generalized sampling series, J. Approx. Theory, 50 (1987), 25-39. doi: 10.1016/0021-9045(87)90063-3.
[11]	L. Coroianu, D. Costarelli, S. G. Gal and G. Vinti, The max-product generalized sampling operators: convergence and quantitative estimates, Appl. Math. Comput., 355 (2019), 173-183. doi: 10.1016/j.amc.2019.02.076.
[12]	L. Coroianu, D. Costarelli, S. G. Gal and G. Vinti, Approximation by max-product sampling Kantorovich operators with generalized kernels, Anal. Appl., 2019, in press. doi: 10.1142/S0219530519500155.
[13]	L. Coroianu and S. G. Gal, Approximation by nonlinear generalized sampling operators of max-product kind, Sampl. Theory Signal Image Process., 9 (2010), 59-75.
[14]	L. Coroianu and S. G. Gal, Approximation by max-product sampling operators based on sinc-type kernels, Sampl. Theory Signal Image Process., 10 (2011), 211-230.
[15]	L. Coroianu and S. G. Gal, Classes of functions with improved estimates in approximation by the max-product Bernstein operator, Anal. Appl., 9 (2011), 249-274. doi: 10.1142/S0219530511001856.
[16]	L. Coroianu and S. G. Gal, Approximation by truncated max-product operators of Kantorovich-type based on generalized $(\varphi, \psi)$-kernels, Math. Meth. Appl. Sci., 41 (2018), 7971-7984. doi: 10.1002/mma.5262.
[17]	L. Coroianu and S. G. Gal, $L^{p}$-approximation by truncated max-product sampling operators of Kantorovich-type based on Fejér kernel, J. Integr. Equ. Appl., 29 (2017), 349-364. doi: 10.1216/JIE-2017-29-2-349.
[18]	L. Coroianu and S. G. Gal, Saturation results for the truncated max-product sampling operators based on sinc and Fejér-type kernels, Sampl. Theory Signal Image Process., 11 (2012), 113-132.
[19]	D. Costarelli, A. M. Minotti and G. Vinti, Approximation of discontinuous signals by sampling Kantorovich series, J. Math. Anal. Appl., 450 (2017), 1083-1103. doi: 10.1016/j.jmaa.2017.01.066.
[20]	D. Costarelli and A. R. Sambucini, Approximation results in Orlicz spaces for sequences of Kantorovich max-product neural network operators, Results Math., 73 (2018), Art. 15. doi: 10.1007/s00025-018-0799-4.
[21]	D. Costarelli, A. R. Sambucini and G. Vinti, Convergence in Orlicz spaces by means of the multivariate max-product neural network operators of the Kantorovich type and applications, Neural Comput. Appl., 31 (2019), 5069-5078. doi: 10.1007/s00521-018-03998-6.
[22]	D. Costarelli and G. Vinti, Order of approximation for sampling Kantorovich type operators, J. Integr. Equ. Appl., 26 (2014), 345-368. doi: 10.1216/JIE-2014-26-3-345.
[23]	D. Costarelli and G. Vinti, Rate of approximation for multivariate sampling Kantorovich operators on some functions spaces, J. Integr. Equ. Appl., 26 (2014), 455-481. doi: 10.1216/JIE-2014-26-4-455.
[24]	D. Costarelli and G. Vinti, Approximation by max-product neural network operators of Kantorovich type, Results Math., 69 (2016), 505-519. doi: 10.1007/s00025-016-0546-7.
[25]	D. Costarelli and G. Vinti, Max-product neural network and quasi-interpolation operators activated by sigmoidal functions, J. Approx. Theory, 209 (2016), 1-22. doi: 10.1016/j.jat.2016.05.001.
[26]	D. Costarelli and G. Vinti, Pointwise and uniform approximation by multivariate neural network operators of the max-product type, Neural Netw., 81 (2016), 81-90.
[27]	D. Costarelli and G. Vinti, Convergence results for a family of Kantorovich max-product neural network operators in a multivariate setting, Math. Slovaca, 67 (2017), 1469-1480. doi: 10.1515/ms-2017-0063.
[28]	D. Costarelli and G. Vinti, Convergence for a family of neural network operators in Orlicz spaces, Math. Nachr., 290 (2017), 226-235. doi: 10.1002/mana.201600006.
[29]	D. Costarelli and G. Vinti, Estimates for the neural network operators of the max-product type with continuous and $p$-integrable functions, Results Math., 73 (2018), Art. 12. doi: 10.1007/s00025-018-0790-0.
[30]	D. Costarelli and G. Vinti, An inverse result of approximation by sampling Kantorovich series, Proc. Edinb. Math. Soc., 62 (2019), 265-280. doi: 10.1017/s0013091518000342.
[31]	S. Y. Güngör and N. Ispir, Approximation by Bernstein-Chlodowsky operators of max-product kind, Math. Commun., 23 (2018), 205-225.
[32]	A. Holhos, Weighted Approximation of functions by Meyer-König and Zeller operators of max-product type, Numer. Funct. Anal. Optim., 39 (2018), 689-703. doi: 10.1080/01630563.2017.1413386.
[33]	A. Holhos, Weighted approximation of functions by Favard operators of max-product type, Period. Math. Hungar., 77 (2018), 340-346. doi: 10.1007/s10998-018-0249-9.
[34]	B. Z. Li, Approximation by multivariate Bernstein-Durrmeyer operators and learning rates of least-squares regularized regression with multivariate polynomial kernels, J. Approx. Theory, 173 (2013), 33-55. doi: 10.1016/j.jat.2013.04.007.
[35]	O. Orlova and G. Tamberg, On approximation properties of generalized Kantorovich-type sampling operators, J. Approx. Theory, 201 (2016), 73-86. doi: 10.1016/j.jat.2015.10.001.
[36]	R. J. Ravier and R. S. Stichartz, Sampling theory with average values on the Sierpinski gasket, Constr. Approx., 44 (2016), 159-194. doi: 10.1007/s00365-016-9341-7.
[37]	R. L. Stens, Error estimates for sampling sums based on convolution integrals, Inform. Control, 45 (1980), 37-47. doi: 10.1016/S0019-9958(80)90857-8.
[38]	D. X. Zhou, Deep distributed convolutional neural networks: universality, Anal. Appl., 16 (2018), 895-919. doi: 10.1142/S0219530518500124.
[39]	D. X. Zhou, Universality of deep convolutional neural networks, Appl. Comput. Harmon. Anal., 48 (2020), 787-794. doi: 10.1016/j.acha.2019.06.004.
[40]	D. X. Zhou and K. Jetter, Approximation with polynomial kernels and SVM classifiers, Adv. Comput. Math., 25 (2006), 323-344. doi: 10.1007/s10444-004-7206-2.

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American Institute of Mathematical Sciences

Approximation by multivariate max-product Kantorovich-type operators and learning rates of least-squares regularized regression

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