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arxiv: 2306.06213 · v3 · submitted 2023-06-09 · 💻 cs.LG · math.OC

A Robust Twin Parametric Margin Support Vector Machine for Multiclass Classification

Renato De Leone , Francesca Maggioni , Andrea Spinelli This is my paper

Pith reviewed 2026-05-24 08:23 UTC · model grok-4.3

classification 💻 cs.LG math.OC
keywords robust optimizationtwin parametric margin SVMmulticlass classificationfeature uncertaintysupport vector machineskernel methodsrobust classificationparametric margin
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The pith

Robust twin parametric margin support vector machines classify multiclass data with feature uncertainty.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces new Twin Parametric Margin Support Vector Machine models for multiclass classification that account for uncertainty in the features of training data. It uses robust optimization to create versions of these models that remain effective even when observations are perturbed within bounded sets. Both linear and kernel-based versions are provided with tractable mathematical reformulations, along with two options for how to combine the classifiers into a final decision. The approach is tested on real datasets to show it handles uncertainty well.

Core claim

The authors derive robust counterparts of the deterministic TPMSVM models by constructing norm-bounded uncertainty sets around each training observation and applying robust optimization techniques, resulting in computationally tractable problems for both linear and kernel-induced classifiers in the multiclass setting.

What carries the argument

Robust TPMSVM models obtained by replacing the deterministic constraints with their robust counterparts under norm-bounded uncertainty sets, solved via robust optimization for linear and kernel cases.

If this is right

  • Tractable reformulations exist for both linear and kernel versions of the robust multiclass TPMSVM.
  • Two alternative decision functions are available to combine the classifiers.
  • The models demonstrate good performance on real-world datasets when feature uncertainty is present.
  • Uncertainty is modeled using bounded-by-norm sets around each observation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The method could be applied to other parametric margin SVM variants beyond twin models.
  • Performance gains might be larger in domains with high measurement noise such as sensor data.
  • Further work could compare these robust models against other uncertainty-handling techniques like distributionally robust optimization.

Load-bearing premise

Bounded-by-norm uncertainty sets around each training observation suffice to represent feature uncertainty and produce tractable robust optimization problems.

What would settle it

Running the proposed robust TPMSVM on a dataset with known feature perturbations where it fails to maintain classification accuracy compared to the non-robust version, or where the reformulated problems cannot be solved efficiently.

Figures

Figures reproduced from arXiv: 2306.06213 by Andrea Spinelli, Francesca Maggioni, Renato De Leone.

Figure 1
Figure 1. Figure 1: Scheme of the selected TWSVM literature review. Th [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Linear and nonlinear classifiers for the case of bin [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Linear and nonlinear classifiers for the case of thr [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
read the original abstract

In this paper, we introduce novel Twin Parametric Margin Support Vector Machine (TPMSVM) models designed to address multiclass classification tasks under feature uncertainty. To handle data perturbations, we construct bounded-by-norm uncertainty set around each training observation and derive the robust counterparts of the deterministic models using robust optimization techniques. To capture complex data structure, we explore both linear and kernel-induced classifiers, providing computationally tractable reformulations of the resulting robust models. Additionally, we propose two alternatives for the final decision function, enhancing models' flexibility. Finally, we validate the effectiveness of the proposed robust multiclass TPMSVM methodology on real-world datasets, showing the good performance of the approach in the presence of uncertainty.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper introduces robust Twin Parametric Margin Support Vector Machine (TPMSVM) formulations for multiclass classification under feature uncertainty. It constructs norm-bounded uncertainty sets around training points, derives robust counterparts of the deterministic TPMSVM models via robust optimization, supplies tractable reformulations for both linear and kernel cases, proposes two alternative decision functions, and reports empirical performance on real-world datasets.

Significance. If the claimed tractable reformulations hold and the empirical gains are reproducible, the work would extend robust optimization techniques to the twin-margin multiclass setting, offering a concrete alternative to standard robust SVMs when data perturbations are modeled by norm balls.

major comments (2)
  1. [§4] §4 (Kernel case): the abstract and introduction assert that the robust kernel TPMSVM admits a computationally tractable reformulation, yet no explicit dual derivation or complexity argument is supplied showing that the semi-infinite program remains a convex QP or SOCP once the twin parametric margins and the one-vs-one/one-vs-rest multiclass constraints are incorporated; the implicit feature map makes preservation of finite-dimensional convexity non-obvious.
  2. [§3.2] §3.2, Eq. (robust counterpart): the reduction of the robust linear model to a finite program is presented without an intermediate step verifying that the worst-case perturbation over the norm ball can be replaced by a dual-norm term without introducing additional non-convexity from the parametric margin formulation.
minor comments (2)
  1. Notation for the two proposed decision functions is introduced without a clear comparison table showing their computational cost versus accuracy trade-off on the reported datasets.
  2. The experimental section reports “good performance” but does not include statistical significance tests or comparison against recent robust multiclass baselines (e.g., robust one-vs-rest SVMs with the same uncertainty set).

Simulated Author's Rebuttal

2 responses · 0 unresolved

Thank you for the constructive comments. We address each major point below and will revise the manuscript to supply the requested explicit derivations and verification steps.

read point-by-point responses

  1. Referee: [§4] §4 (Kernel case): the abstract and introduction assert that the robust kernel TPMSVM admits a computationally tractable reformulation, yet no explicit dual derivation or complexity argument is supplied showing that the semi-infinite program remains a convex QP or SOCP once the twin parametric margins and the one-vs-one/one-vs-rest multiclass constraints are incorporated; the implicit feature map makes preservation of finite-dimensional convexity non-obvious.

    Authors: We agree that the kernel-case section would be strengthened by an explicit dual derivation. In the revision we will add the full dual formulation for the robust kernel TPMSVM, first replacing the semi-infinite robust constraints by their dual-norm equivalents and then applying the kernel trick; the resulting program remains a convex QP because the twin parametric margin terms stay linear in the dual variables and the kernel matrix is positive semi-definite. revision: yes

  2. Referee: [§3.2] §3.2, Eq. (robust counterpart): the reduction of the robust linear model to a finite program is presented without an intermediate step verifying that the worst-case perturbation over the norm ball can be replaced by a dual-norm term without introducing additional non-convexity from the parametric margin formulation.

    Authors: The observation is correct; an intermediate verification step is missing. We will insert this step in the revision, explicitly showing that the worst-case term over the norm ball is replaced by its dual-norm expression while the parametric margin constraints remain linear (hence convex) in the decision variables. revision: yes

Circularity Check

0 steps flagged

No circularity: robust reformulations derived from standard techniques

full rationale

The paper starts from deterministic TPMSVM formulations, applies bounded-norm uncertainty sets, and derives robust counterparts via robust optimization for both linear and kernel cases. No equations or claims reduce a result to its own inputs by definition, no fitted parameters are relabeled as predictions, and no load-bearing steps rely on self-citations whose content is unverified. The tractability of the reformulations is presented as an output of the derivation rather than an input assumption that forces the conclusion.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based solely on the abstract, no explicit free parameters, axioms, or invented entities are detailed; the work relies on standard robust optimization applied to existing SVM frameworks.

pith-pipeline@v0.9.0 · 5642 in / 1112 out tokens · 22471 ms · 2026-05-24T08:23:49.939963+00:00 · methodology

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Works this paper leans on

72 extracted references · 72 canonical work pages

  1. [1]

    Angulo, C., Parra, X., Catal` a, A., 2003. K-svcr. a support vect or machine for multi-class classification. Neurocomputing 55, 57–77

    work page 2003

  2. [2]

    Least squares twin support vec tor machines for pattern classification

    Arun Kumar, M., Gopal, M., 2009. Least squares twin support vec tor machines for pattern classification. Expert Systems with Applications 36, 7535–43

    work page 2009

  3. [3]

    Robust optimizatio n, Princeton University Press

    Ben-Tal, A., El Ghaoui, L., Nemirovski, A., 2009. Robust optimizatio n, Princeton University Press

    work page 2009

  4. [4]

    Robust cla ssification

    Bertsimas, D., Dunn, J., Pawlowski, C., Zhuo, Y.D., 2019. Robust cla ssification. INFORMS Journal of Optimization 1, 2–34. 26

    work page 2019

  5. [5]

    On lp-suppo rt vector machines and multidimensional kernels

    Blanco, V., Puerto, J., Rodr ´ ıguez-Ch ´ ıa, A.M., 2020. On lp-suppo rt vector machines and multidimensional kernels. Journal of Machine Learning Research 21 , 1–29

    work page 2020

  6. [6]

    A training algorithm for o ptimal margin classifiers

    Boser, B.E., Guyon, I., Vapnik, V.N., 1992. A training algorithm for o ptimal margin classifiers. Proceedings of the Fifth Annual Workshop of Computational Lear ning Theory 5, 144–52

    work page 1992

  7. [7]

    Multicategory classific ation by support vector ma- chines

    Bredensteiner, E.J., Bennett, K.P., 1999. Multicategory classific ation by support vector ma- chines. Computational Optimization and Applications 12, 53–79

    work page 1999

  8. [8]

    Clu stering categories in support vector machines

    Carrizosa, E., Nogales-G´ omez, A., Romero Morales, D., 2017. Clu stering categories in support vector machines. Omega 66, 28–37

    work page 2017

  9. [9]

    Chamasemani, F.F., Singh, Y.P., 2011. Multi-class support vector m achine (svm) classifiers – an application in hypothyroid detection and classification, in: 2011 S ixth International Conference on Bio-Inspired Computing: Theories and Applications

    work page 2011

  10. [10]

    A new fuzzy twin support vector machin e for pattern classification

    Chen, S.G., Wu, X., 2018. A new fuzzy twin support vector machin e for pattern classification. International Journal of Machine Learning and Cybernetics 9, 15 53–64

    work page 2018

  11. [11]

    Proportional sampling strat egy: a compendium and some insights

    Chen, T.Y., Tse, T.H., Yu, Y.T., 2001. Proportional sampling strat egy: a compendium and some insights. The Journal of Systems and Software 58, 65–81

    work page 2001

  12. [12]

    Recursive projection twin support vector machine via within-class variance minimization

    Chen, X., Yang, J., Ye, Q., Liang, J., 2011. Recursive projection twin support vector machine via within-class variance minimization. Pattern Recognition 44, 2643– 55

    work page 2011

  13. [13]

    Support-vector networks

    Cortes, C., Vapnik, V.N., 1995. Support-vector networks. Ma chine Learning 20, 273–97

    work page 1995

  14. [14]

    Support vector regression for time series analysis, in: Operations Research, Springer

    De Leone, R., 2010. Support vector regression for time series analysis, in: Operations Research, Springer

    work page 2010

  15. [15]

    De Leone, R., Maggioni, F., Spinelli, A., 2024. A multiclass robust twin parametric mar- gin support vector machine with an application to vehicles emissions, in : Nicosia, G., Ojha, V., La Malfa, E., La Malfa, G., Pardalos, P.M., Umeton, R. (Eds.), Machine Learn- ing, Optimization, and Data Science, Springer Nature Switzerland, C ham. pp. 299–310. doi:ht...

    work page doi:10.1007/978-3-031-53966-4_22 2024

  16. [16]

    A review on multi-class twsvm

    Ding, S., Zhao, X., Zhang, J., Zhang, X., Xue, Y., 2019. A review on multi-class twsvm. Artificial Intelligence Review 52, 775–801

    work page 2019

  17. [17]

    A multiclass nonparallel parametric-margin support vector machine

    Du, S.W., Zhang, M.C., Chen, P., Sun, H.F., Chen, W.J., Shao, Y.H., 202 1. A multiclass nonparallel parametric-margin support vector machine. Informa tion 12, 515–33. 27

    work page

  18. [18]

    Robust and distributio nally robust optimization models for linear support vector machine

    Faccini, D., Maggioni, F., Potra, F.A., 2022. Robust and distributio nally robust optimization models for linear support vector machine. Computers and Operatio ns Research 147, 105930

    work page 2022

  19. [19]

    Using machine lear ning techniques to reduce uncertainty for outpatient appointment scheduling practices in ou tpatient clinics

    Golmohammadi, D., Zhao, L., Dreyfus, D., 2023. Using machine lear ning techniques to reduce uncertainty for outpatient appointment scheduling practices in ou tpatient clinics. Omega 120, 102907

    work page 2023

  20. [20]

    Graph implementations for nonsmoot h convex programs, in: Blondel, V., Boyd, S., Kimura, H

    Grant, M., Boyd, S., 2008. Graph implementations for nonsmoot h convex programs, in: Blondel, V., Boyd, S., Kimura, H. (Eds.), Recent Advances in Learning and Control. Springer-Verlag Limited. Lecture Notes in Control and Informatio n Sciences, pp. 95–110. http://stanford.edu/~boyd/graph_dcp.html

    work page 2008

  21. [21]

    CVX: Matlab software for disciplined c onvex programming, version 2.1

    Grant, M., Boyd, S., 2014. CVX: Matlab software for disciplined c onvex programming, version 2.1. http://cvxr.com/cvx

    work page 2014

  22. [22]

    Deep learning for credit scoring: Do or don’t? European Journal of Operational Research 295, 292–305

    Gunnarsson, B.R., vanden Broucke, S., Baesens, B., ´Oskarsd´ ottir, M., Lemahieu, W., 2021. Deep learning for credit scoring: Do or don’t? European Journal of Operational Research 295, 292–305

    work page 2021

  23. [23]

    New support vector algorithms with parametric insensitive/margin model

    Hao, P.Y., 2010. New support vector algorithms with parametric insensitive/margin model. Neural networks : the official journal of the International Neur al Network Society 23, 60–73

    work page 2010

  24. [24]

    A comparison of methods for multiclass s upport vector machines

    Hsu, C.W., Lin, C.J., 2002. A comparison of methods for multiclass s upport vector machines. IEEE Transactions on Neural Networks 13, 415–25

    work page 2002

  25. [25]

    Twin support vector machines for pattern classification

    Jayadeva, Khemchandani, R., Chandra, S., 2007. Twin support vector machines for pattern classification. IEEE Transactions on Pattern Analysis and Machine I ntelligence 29, 905–10

    work page 2007

  26. [26]

    A novel e mbedded min-max approach for feature selection in nonlinear support vector machine classifica tion

    Jim´ enez-Cordero, A., Morales, J.M., Pineda, S., 2021. A novel e mbedded min-max approach for feature selection in nonlinear support vector machine classifica tion. European Journal of Operational Research 293, 24–35

    work page 2021

  27. [27]

    UCI machine learnin g repository

    Kelly, M., Longjohn, R., Nottingham, K., 2023. UCI machine learnin g repository. URL: http://archive.ics.uci.edu/ml

    work page 2023

  28. [28]

    Multi-class svm classifier based on p airwise coupling, in: Lee, S.W., Verri, A

    Li, Z., Tang, S., Yan, S., 2002. Multi-class svm classifier based on p airwise coupling, in: Lee, S.W., Verri, A. (Eds.), Pattern Recognition with Support Vector Mac hines, Springer Berlin Heidelberg, Berlin, Heidelberg. pp. 321–33

    work page 2002

  29. [29]

    Pattern separation and prediction via linear and semidefinite programming

    Liu, X., Potra, F.A., 2009. Pattern separation and prediction via linear and semidefinite programming. Studies in Informatics and Control 18, 71–82. 28

    work page 2009

  30. [30]

    A robust formu lation for twin multiclass support vector machine

    L´ opez, J., Maldonado, S., Carrasco, M., 2017. A robust formu lation for twin multiclass support vector machine. Applied Intelligence 47, 1031–43

    work page 2017

  31. [31]

    Robust nonpar allel support vector machines via second-order cone programming

    L´ opez, J., Maldonado, S., Carrasco, M., 2019. Robust nonpar allel support vector machines via second-order cone programming. Neurocomputing 364, 227–3 8

    work page 2019

  32. [32]

    Ma chine learning based classification models for covid-19 patients, in: Aringhieri, R., Maggion i, F., Lanzarone, E., Reuter-Oppermann, M., Righini, G., Vespucci, M.T

    Maggioni, F., Faccini, D., Gheza, F., Manelli, F., Bonetti, G., 2023. Ma chine learning based classification models for covid-19 patients, in: Aringhieri, R., Maggion i, F., Lanzarone, E., Reuter-Oppermann, M., Righini, G., Vespucci, M.T. (Eds.), Operation s Research for Health Care in Red Zone, Springer International Publishing, Cham. pp. 35– 46

    work page 2023

  33. [33]

    A robust optimization model for non linear support vector machine

    Maggioni, F., Spinelli, A., 2023. A robust optimization model for non linear support vector machine. arXiv:2306.06223

    work page arXiv 2023

  34. [34]

    A robust nonlinear support vector machine approach for vehicles smog rating classification, in: Bruglieri, M., Festa, P., Mac rina, G., Pisacane, O

    Maggioni, F., Spinelli, A., 2024. A robust nonlinear support vector machine approach for vehicles smog rating classification, in: Bruglieri, M., Festa, P., Mac rina, G., Pisacane, O. (Eds.), Optimization in Green Sustainability and Ecological Transitio n, Springer Cham. doi:https://doi.org/10.1007/978-3-031-47686-0_19

    work page doi:10.1007/978-3-031-47686-0_19 2024

  35. [35]

    Pro fit-driven churn prediction for the mutual fund industry: A multisegment approach

    Maldonado, S., Dom ´ ınguez, G., Olaya, D., Verbeke, W., 2021. Pro fit-driven churn prediction for the mutual fund industry: A multisegment approach. Omega 10 0, 102380

    work page 2021

  36. [36]

    A second-orde r cone programming formulation for twin support vector machines

    Maldonado, S., L´ opez, J., Carrasco, M., 2016. A second-orde r cone programming formulation for twin support vector machines. Applied Intelligence 45, 265–76

    work page 2016

  37. [37]

    The cobb-doug las learning machine

    Maldonado, S., L´ opez, J., Carrasco, M., 2022. The cobb-doug las learning machine. Pattern Recognition 128, 108701

    work page 2022

  38. [38]

    Profit-based chu rn prediction based on minimax probability machines

    Maldonado, S., L´ opez, J., Vairetti, C., 2020. Profit-based chu rn prediction based on minimax probability machines. European Journal of Operational Research 284, 273–84

    work page 2020

  39. [39]

    Multiclass skin lesion loca lization and classification using deep learning based features fusion and selection framework for smart healthcare

    Maqsood, S., Damaˇ seviˇ cius, R., 2023. Multiclass skin lesion loca lization and classification using deep learning based features fusion and selection framework for smart healthcare. Neural Networks 160, 238–58

    work page 2023

  40. [40]

    Multi-kernel covariance terms in multi-output support vector machines, in: Nicosia, G., Ojha, V., La Malfa, E., Jansen, G., Sc iacca, V., Pardalos, P., Giuffrida, G., Umeton, R

    Marcelli, E., De Leone, R., 2020. Multi-kernel covariance terms in multi-output support vector machines, in: Nicosia, G., Ojha, V., La Malfa, E., Jansen, G., Sc iacca, V., Pardalos, P., Giuffrida, G., Umeton, R. (Eds.), Machine Learning, Optimization, a nd Data Science, Springer International Publishing, Cham. pp. 1–11

    work page 2020

  41. [41]

    The MOSEK optimization toolbox for MATLAB ma nual

    MOSEK ApS, 2019. The MOSEK optimization toolbox for MATLAB ma nual. Version 9.1. URL: http://docs.mosek.com/9.1/toolbox/index.html. 29

    work page 2019

  42. [42]

    URL: https://open.canada.ca/en/open-data

    Open Data - Government of Canada, 2023. URL: https://open.canada.ca/en/open-data. Accessed on 05.03.2023

    work page 2023

  43. [43]

    Tpmsvm: A novel twin parametric-margin suppo rt vector machine for pattern recognition

    Peng, X., 2011. Tpmsvm: A novel twin parametric-margin suppo rt vector machine for pattern recognition. Pattern Recognition 44, 2678–92

    work page 2011

  44. [44]

    Improvements on twin param etric-margin support vector machine

    Peng, X., Kong, L., Chen, D., 2015. Improvements on twin param etric-margin support vector machine. Neurocomputing 151, 857–63

    work page 2015

  45. [45]

    Structural twin parametric-m argin support vector machine for binary classification

    Peng, X., Wang, Y., Xu, D., 2013. Structural twin parametric-m argin support vector machine for binary classification. Knowledge-Based Systems 49, 63–72

    work page 2013

  46. [46]

    Robust minimum class variance twin suppor t vector machine classifier

    Peng, X., Xu, D., 2013. Robust minimum class variance twin suppor t vector machine classifier. Neural Computing and Applications 22, 999–1011

    work page 2013

  47. [47]

    Famcdm: A fusion appro ach of mcdm methods to rank multiclass classification algorithms

    Peng, Y., Kou, G., Wang, G., Shi, Y., 2011. Famcdm: A fusion appro ach of mcdm methods to rank multiclass classification algorithms. Omega 39, 677–89

    work page 2011

  48. [48]

    Large margin da gs for multiclass classifi- cation, in: Solla, S., Leen, T., M¨ uller, K

    Platt, J., Cristianini, N., Shawe-Taylor, J., 1999. Large margin da gs for multiclass classifi- cation, in: Solla, S., Leen, T., M¨ uller, K. (Eds.), Advances in Neural I nformation Processing Systems, MIT Press. pp. 547–53

    work page 1999

  49. [49]

    Robust twin support vector machine for pattern classification

    Qi, Z., Tian, Y., Shi, Y., 2013. Robust twin support vector machine for pattern classification. Pattern Recognition 46, 305–16

    work page 2013

  50. [50]

    Frameworks and results in dist ributionally robust opti- mization

    Rahimian, H., Mehrotra, S., 2022. Frameworks and results in dist ributionally robust opti- mization. Open Journal of Mathematical Optimization 3, 1–85

    work page 2022

  51. [51]

    Real and complex analysis

    Rudin, W., 1987. Real and complex analysis. McGraw-Hill

    work page 1987

  52. [52]

    Improved robust nonparallel suppo rt vector machines

    Sahleh, A., Salahi, M., 2022. Improved robust nonparallel suppo rt vector machines. Interna- tional Journal of Data Science and Analytics 1, 1–14

    work page 2022

  53. [53]

    New support vector algo- rithms

    Sch¨ olkopf, B., Smola, A., Williamson, R.C., Bartlett, P.L., 2000. New support vector algo- rithms. Neural Computation 12, 1207–45

    work page 2000

  54. [54]

    Learning with Kernels: Suppor t Vector Machines, regular- ization, optimization, and beyond

    Sch¨ olkopf, B., Smola, A.J., 2001. Learning with Kernels: Suppor t Vector Machines, regular- ization, optimization, and beyond. MIT press

    work page 2001

  55. [55]

    Least squares t win parametric-margin support vector machine for classification

    Shao, Y., Wang, Z., Chen, W.J., Deng, N., 2013a. Least squares t win parametric-margin support vector machine for classification. Applied Intelligence 39, 4 51 –64. 30

    work page

  56. [56]

    T he best separating decision tree twin support vector machine for multi-class classificat ion

    Shao, Y.H., Chen, W.J., Huang, W.B., Yang, Z.M., Deng, N.Y., 2013b. T he best separating decision tree twin support vector machine for multi-class classificat ion. Procedia Computer Science 17, 1032–8

    work page

  57. [57]

    Lectures o n Stochastic Programming - Modeling and Theory

    Shapiro, A., Dentcheva, D., Ruszczynski, A., 2009. Lectures o n Stochastic Programming - Modeling and Theory. MOS-SIAM Series on Optimization

    work page 2009

  58. [58]

    Comprehen sive review on twin support vector machines

    Tanveer, M., Rajani, T., Rastogi, R., Shao, Y., 2022. Comprehen sive review on twin support vector machines. Annals of Operations Research 310, 1–46

    work page 2022

  59. [59]

    Robust classification and regre ssion using support vector machines

    Trafalis, T.B., Gilbert, R.C., 2006. Robust classification and regre ssion using support vector machines. European Journal of Operational Research 173, 893– 909

    work page 2006

  60. [60]

    Suppo rt vector frontiers: A new approach for estimating production functions through support v ector machines

    Valero-Carreras, D., Aparicio, J., Guerrero, N.M., 2021. Suppo rt vector frontiers: A new approach for estimating production functions through support v ector machines. Omega 104, 102490

    work page 2021

  61. [61]

    Theory of Pattern Reco gnition

    Vapnik, V.N., Chervonenkis, A.Y., 1974. Theory of Pattern Reco gnition. Nauka, Moscow

    work page 1974

  62. [62]

    Twin-parametric margin suppor t vector machine with truncated pinball loss

    Wang, H., Xu, Y., Zhou, Z., 2021. Twin-parametric margin suppor t vector machine with truncated pinball loss. Neural Computing and Applications 33, 3781 –98

    work page 2021

  63. [63]

    A ga-based model selection f or smooth twin parametric- margin support vector machine

    Wang, Z., Shao, Y.H., Wu, T.R., 2013. A ga-based model selection f or smooth twin parametric- margin support vector machine. Pattern Recognition 46, 2267–77

    work page 2013

  64. [64]

    Extending t win support vector machine classifier for multi-category classification problems

    Xie, J., Hone, K.S., Xie, W., Gao, X., Shi, Y., Liu, X., 2013. Extending t win support vector machine classifier for multi-category classification problems. Intellig ent Data Analysis 17, 649–64

    work page 2013

  65. [65]

    Robustness and regula rization of support vector machines

    Xu, H., Caramanis, C., Mannor, S., 2009. Robustness and regula rization of support vector machines. Journal of Machine Learning Research 10, 1485–510

    work page 2009

  66. [66]

    A twin multi-class classification sup port vector machine

    Xu, Y., Guo, R., Wang, L., 2013. A twin multi-class classification sup port vector machine. Cognitive Computation 5, 580–8

    work page 2013

  67. [67]

    A novel twin support-vector ma chine with pinball loss

    Xu, Y., Yang, Z., Pan, X., 2017. A novel twin support-vector ma chine with pinball loss. IEEE Transactions on Neural Networks and Learning Systems 28, 359– 70

    work page 2017

  68. [68]

    Linear programming approaches for multicate gory support vector machines

    Yajima, Y., 2005. Linear programming approaches for multicate gory support vector machines. European Journal of Operational Research 162, 514–31

    work page 2005

  69. [69]

    Multiple birth support vect or machine for multi-class classification

    Yang, Z., Shao, Y., Zhang, X.S., 2013. Multiple birth support vect or machine for multi-class classification. Neural Computing and Applications 22, 153–61. 31

    work page 2013

  70. [70]

    Hyper-parameter optimization: A review o f algorithms and applica- tions

    Yu, T., Zhu, H., 2020. Hyper-parameter optimization: A review o f algorithms and applica- tions. arXiv:2003.05689

    work page arXiv 2020

  71. [71]

    Second-order cone programm ing formulations for robust multiclass classification

    Zhong, P., Fukushima, M., 2007. Second-order cone programm ing formulations for robust multiclass classification. Neural Computation 19, 258–82

    work page 2007

  72. [72]

    Big data and por tfolio optimization: A novel approach integrating dea with multiple data sources

    Zhou, Z., Gao, M., Xiao, H., Wang, R., Liu, W., 2021. Big data and por tfolio optimization: A novel approach integrating dea with multiple data sources. Omega 104, 102479. 32

    work page 2021