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Stochastic AUC optimization with general loss
Approximation by multivariate max-product Kantorovich-type operators and learning rates of least-squares regularized regression
1. | Department of Mathematics and Computer Science, University of Oradea, Oradea, Romania |
2. | Department of Mathematics and Computer Science, University of Perugia, Perugia, Italy |
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.
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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