GBTSVR: Twin Support Vector Regression via Granular Ball Computing

A. Quadir; M. Tanveer

cs.LGOct 14, 2026

GBTSVR: Twin Support Vector Regression via Granular Ball Computing

Authors: A. Quadir, M. Tanveer

Organizations: Department of Mathematics, Indian Institute of Technology Indore, Simrol, Indore, 453552, Madhya Pradesh, India

Abstract

Regression models used in real-world applications are often challenged by noise, outliers, data heterogeneity, and asymmetric error characteristics, which severely limit the effectiveness of conventional point-wise learning models. While twin support vector regression (TSVR) partially addresses these issues through the use of nonparallel regression bounds, it still relies on individual samples as basic learning units. As a result, TSVR remains highly sensitive to noisy and contaminated observations and fails to leverage the underlying structural and regional information inherent in complex datasets, leading to suboptimal robustness and generalization performance. To address these limitations, we introduce granular ball twin support vector regression (GBTSVR) and its quantile extension, granular ball twin support vector quantile regression (GB-TSVQR). The proposed models integrate granular ball computing with TSVR, replacing individual data points with compact, structure-preserving granular balls as fundamental learning units. This region-level representation effectively suppresses noise and outliers while preserving intrinsic data geometry, thereby enhancing robustness, stability, and generalization. GBTSVR constructs nonparallel lower- and upper-bound regression functions over granular balls, significantly reducing sensitivity to sample-level perturbations. Building upon this framework, GB-TSVQR further incorporates quantile regression principles to simultaneously capture heterogeneous trends and asymmetric error distributions across different quantile levels. Extensive experiments on synthetic datasets, time-series benchmarks, and outlier-contaminated data demonstrate that the proposed GBTSVR and GB-TSVQR consistently outperform state-of-the-art regression models.

Figures & tables

Panel of Figure 1 — Figure 1: This figure demonstrates the construction of granular balls by segmenting the sinc dataset corrupted with Gaussian noise of zero mean and a standard deviation of $0.2$ .

Figure 2: Overview of the GBTSVR Framework.

\widetilde{\mathcal{G}}=\mathcal{G}\mathcal{G}^{T},

Algorithm 1 Dual Coordinate Descent Method (DCDM)

Dataset	Metric	SVR [ 5 ]	TSVR [ 17 ]	TSVQR [ 21 ]	$L_{1}$ -TSVQR [ 33 ]	GL-TPSVR [ 34 ]	GBTSVR †	GB-TSVQR †
2014_2015 CSM dataset	RMSE	0.37972	1.33951	1.23384	0.40545	0.44613	0.00755	0.00065
$(185\times 11)$	MAE	0.22234	0.77661	0.79133	0.39927	0.35916	0.00366	0.00057
	Pos Error	0.43571	0.96693	1.20578	0.9502	0.66887	0.00601	0.00058
	Neg Error	0.17018	0.45941	0.50247	0.01384	0.3288	0.00213	0.00006
Abalone	RMSE	0.0062	0.38768	0.07667	0.00656	0.45811	0.00265	0.00073
$(4177\times 7)$	MAE	0.00448	0.2541	0.07666	0.00652	0.42865	0.00216	0.00062

Table 1: Performance comparison in terms of RMSE, MAE, Pos Error, and Neg Error on benchmark UCI datasets for the proposed GBTSVR and GB-TSVQR models with baseline models.

Dataset	Generating Function	Input Domain
Function 1	$y_{i}=sin(x_{i})+\epsilon_{i}$	$x_{i}\in[0,2\pi]$
Function 2	$y_{i}=\frac{sin(3x_{i})}{3x_{i}}+\epsilon_{i}$	$x_{i}\in[-4,4]$
Function 3	$y_{i}=sin(x_{i})cos(x_{i}^{2})+\epsilon_{i}$	$x_{i}\in[0,2\pi]$
Function 4	$y_{i}=x_{i}cos(x_{i})+\epsilon_{i}$	$x_{i}\in[-4,4]$
Function 5	$y_{i}=(2x_{i}^{2}-x_{i}+1)\exp(-\frac{x_{i}^{2}}{2})+\epsilon_{i}$	$x_{i}\in[-4,4]$
The noise term $\epsilon_{i}$ follows different distributions depending on the dataset:

Table 2: Overview of the synthetic datasets used in the experiments.

Dataset	Metric	SVR [ 5 ]	TSVR [ 17 ]	TSVQR [ 21 ]	$L_{1}$ -TSVQR [ 33 ]	GL-TPSVR [ 34 ]	GBTSVR †	GB-TSVQR †
Type 1: Gaussian noise with zero mean and a standard deviation of 0.2.
Function 1	RMSE	0.98218	0.65454	0.65441	0.98456	0.70711	0.00081	0.00067
Function 2	RMSE	1.34294	0.66864	0.66667	0.95784	0.75896	0.00095	0.00042
Function 3	RMSE	1.12861	0.78956	0.62874	0.85742	0.84528	0.00085	0.00058
Function 4	RMSE	0.95022	0.74268	0.45929	0.91428	0.71587	0.00074	0.00062
Function 5	RMSE	0.96827	0.79533	0.59533	0.87458	0.70711	0.00082	0.00063

Table 3: RMSE performance on synthetic datasets comparing the proposed GBTSVR and GB-TSVQR models with baseline models.

Dataset	Metric	SVR [ 5 ]	TSVR [ 17 ]	TSVQR [ 21 ]	$L_{1}$ -TSVQR [ 33 ]	GL-TPSVR [ 34 ]	GBTSVR †	GB-TSVQR †
Type 1: Gaussian noise with zero mean and a standard deviation of 0.2.
Function 1	RMSE	1.00601	0.82001	0.82043	0.96129	0.93508	0.00076	0.00018
Function 2	RMSE	0.30138	1.49669	1.49712	0.92229	0.93209	0.94228	0.0204
Function 3	RMSE	0.87983	1.46122	0.31359	0.9801	0.93056	0.00097	0.00018
Function 4	RMSE	1.71677	1.1704	1.16405	0.97147	0.99251	0.94252	0.09715
Function 5	RMSE	0.65021	0.92874	0.92931	0.91363	0.94905	0.92991	0.09109

Table 4: RMSE comparison on synthetic datasets with injected outliers for the proposed GBTSVR and GB-TSVQR models and baseline models.

Dataset	Metric	SVR [ 5 ]	TSVR [ 17 ]	TSVQR [ 21 ]	$L_{1}$ -TSVQR [ 33 ]	GL-TPSVR [ 34 ]	GBTSVR †	GB-TSVQR †
NNGC1_dataset_D1_V1_003	RMSE	0.9938	0.01041	0.81541	0.00875	0.98186	0.00064	0.00073
$(430,5)$	MAE	0.84253	0.00826	0.68345	0.00786	0.83049	0.00053	0.00072
	Pos Error	0.76537	0.00662	0.65806	0.00736	0.86224	0.00059	-
	Neg Error	0.90556	0.01014	0.70846	0.00742	0.802	0.00048	0.00072
NNGC1_dataset_D1_V1_004	RMSE	0.65551	0.00277	0.46681	0.0084	0.65176	0.00033	0.00085
$(545,5)$	MAE	0.54944	0.00241	0.35603	0.0075	0.5423	0.00026	0.00077

Table S.1 : Performance comparison of the proposed GBTSVR and GB-TSVQR models and baseline models on time-series datasets in terms of RMSE, MAE, Poss Error, and Neg Error.

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Université Paris Cité, CNRS, MAP5, F-75006 Paris, France · Université Marie et Louis Pasteur, CNRS, LmB (UMR 6623), F-25000 Besançon, France · Université Paris Cité, Université Paris Saclay, ENS Paris Saclay, CNRS, SSA, INSERM, Centre Borelli, F-75006, Paris, France

Quantile Regression OOD Generalization Support Vector Machines Extreme Value Theory

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We study quantile regression in an extrapolation regime where the covariate takes unusually large values. Under regular variation assumptions, extreme observations can be effectively characterized through their angular components, enabling learning strategies that focus on the angle of the most extreme observations. This approach is formalized through the minimization of an asymptotic conditional risk that localizes learning in the tail of the covariate distribution. We propose a novel Support Vector Machine (SVM) framework for extreme quantile regression, leveraging reproducing kernel Hilbert spaces to handle high-dimensional and nonlinear settings. Our method also accommodates unbounded response variables and avoids restrictive transformations. We establish finite-sample learning guarantees under mild regularity assumptions. The proposed framework unifies ideas from statistical learning and multivariate extremes, providing a tractable and theoretically grounded approach to extrapolation. We complement our theoretical findings with an empirical study on river flow data from the Danube, demonstrating the practical relevance of our methods.

Abstract

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