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% mm \journal{Information Sciences }%
\begin{document}
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\begin{frontmatter}
\title{Generalized Adaptive Huber Loss Driven Robust Twin Support Vector Machine Learning Framework for Pattern Classification }
\author{Tiantian Jiang$^a$}
\author{Bo Wei$^*$$^a$}
\address[label1]{School of Mathematics and Information Sciences, North Minzu University, Yinchuan Ningxia 750021, P.R. China. }

%\cortext[cor1]{Corresponding author}

\cortext[cor2]{Corresponding author}
\ead{weibo\_bmd@163.com}
\author{Guolin Yu$^a$}
\author{Jun Ma$^a$}

\begin{abstract}
Aiming at the problem that the traditional Twin Support Vector Machine (TSVM) is sensitive to noise and outliers, this paper proposes a twin support vector machine model based on generalized adaptive Huber loss function (GAHTSVM). Via dynamically adjusting the robust parameters \(s\), the model can effectively suppress the adverse effects of noisy data points on the decision function, and improve the sparsity of the model by introducing insensitive region design. For non-convex optimization problems, concave-convex process (CCCP) is adopted to avoid quadratic programming problems and significantly improve the efficiency of the algorithm. Experiments show that GAHTSVM performs well in the scenarios where Gaussian noise is added to real-world datasets and outliers are introduced to artificial datasets: it is significantly better than other algorithms in low noise environment; In the high noise environment, it still maintains a leading position, and most datasets are ranked in the top; It performs well on two artificial datasets and is significantly superior to other algorithms.In conclusion, the model effectively overcomes the noise and outlier interference by dynamically adjusting the outlier penalty, and provides an efficient solution to the pattern recognition problem in high noise environment.
\end{abstract}

\begin{keyword}
Generalized adaptive Huber loss; Robustness; Concave-convex procedure
\end{keyword}
\end{frontmatter}
\section{Introduction}\label{s1}
Classification is a basic task in the field of machine learning and pattern recognition, and is widely used in various fields \cite{ref1, ref2, ref3, ref4}. The main goal is to divide a set of samples into disjoint categories, thus efficiently mining the underlying information in the data. However, in the process of data acquisition and transmission, the sample data is inevitably polluted by noise or outliers, which may cause the model to overfit or reduce the generalization ability. Therefore, it is crucial to study how to effectively mitigate the negative impact of noise.

Robust classifier is an effective method to resist adversarial disturbance and reduce noise. Support vector machine (SVM) is one of the most representative classification tools. For a given sample of two classes, SVM achieves efficient class separation by constructing two parallel hyperplanes and maximizing the spacing between these two hyperplanes. Although SVM has become a classic machine learning method because of its solid mathematical theory foundation, excellent classification performance and strong generalization ability, there is still some room for improvement. Firstly, SVM is computationally inefficient when dealing with large-scale datasets, especially in the process of solving quadratic programming problems, which limits its application in the big data environment. Secondly, SVM relies on kernel function for mapping when dealing with high-dimensional nonlinear problems, but for some complex data sets, the selection of kernel function and its computational efficiency are still bottlenecks. As a result, the researchers have proposed several directions for improvement to address these challenges.

A common improvement method is to improve the training efficiency of SVM on large-scale data sets by sparse data representation, optimizing kernel methods and improving the solution algorithm. The traditional C-SVM model \cite{ref5} uses Hinge loss function to make it have the characteristics of sparse classification. However, C-SVM suffers from noise sensitivity and unstable resampling. In order to solve these problems, Huang et al. \cite{ref6} proposed the pinball loss support vector Machine (Pin-SVM) model, it penalizes not only misclassified samples, but also correctly classified samples. Since the pinball loss function considers the quantile distance, the Pin-SVM model shows strong robustness and can effectively reduce the sensitivity to noise. However, compared to C-SVM, Pin-SVM compromises on sparsity. To further improve the efficiency of SVM, Jayadeva et al. \cite{ref7} presented the Twin Support Vector Machine (TSVM) model, which enhanced the flexibility of SVM via introducing two non-parallel hyperplanes to divide training samples. Different from the traditional SVM, which only constructs one decision hyperplane, TSVM constructs two hyperplanes for each class, so that the hyperplanes of positive and negative samples are as close as possible to their respective samples while maintaining a long distance. Compared with the large-scale quadratic programming problem (QPP) that SVM needs to solve, the two QPPS in TSVM are relatively small, and the computational efficiency is significantly improved. Twin Support Vector Machine with Pinball Loss (PinTSVM) \cite{ref8} inherited the strong noise immunity of PinSVM and incorporates the idea of TSVM. It does not require two parallel hyperplanes, so it is more flexible and computationally efficient. Pin-TSVM has good noise immunity by minimizing intra-class scattering and maximizing inter-class distance. Wang et al. \cite{ref9} extended a Twin bounded support vector machine with Capped Pinball loss (CPin-GTBSVM), which has the excellent property of being insensitive to feature and label noise. A two-parameter margin support vector machine with truncated pinball loss (TPin-TSVM) \cite{ref10}, which applied truncated pinball loss to improve the robustness of Pin-TSVM, but its solution speed is very slow. Subsequently, Tanveer et al. \cite{ref11} introduced a sparse pinball twin support vector machine based on insensitive region pinball loss function (SPTSVM). It overcomes the original TSVM is not sensitive to noise, and at the same time ensure TSVM sparse sex and remain as fast calculation speed and the original TSVM. \cite{ref12} reformulated based on not sensitive area pinball loss function of sparse pinball double support vector machine (SPTSVM), the method not only overcomes the traditional TSVM sensitivity to noise, we also ensure the sparsity of the model while maintaining the same computational speed as the original TSVM. In addition, a flexible pinball loss support vector Machine (FP-SVM) \cite{ref13} model for breast cancer prediction has been proposed, which ensures that the convexity of the optimization problem is preserved by providing losses for misclassified and correctly classified samples with parameter sums and strategically traversing the maximum solution path. A SVM combining robust, bounded, sparse, and smooth loss functions (LRoBoss-SVM) \cite{ref14} designed to handle outliers and high-dimensional data while improving the convergence speed in the training process and is applied to supervised learning. Mushir Akhtar et al. \cite{ref15} presented a Wave loss function, which is robust and insensitive to noise, bounded and critical smoothness, and applied it to SVM and TSVM to obtain robust and smooth Wave-SVM and Wave-TSVM models, respectively. Via spectral decomposition and kernel transformation, \cite{ref16} proved that the graph-based method is equivalent to supervised kernel learning, and a spectral kernel design framework is proposed to improve the adaptability of the model to data structures, thereby enhancing the classification performance. \cite{ref17} combined with hyper-heuristic Salp swarm Optimization (HHSSO), the best kernels (including linear, polynomial, Gaussian and RBF kernels) and their corresponding parameters are automatically selected, which effectively reduces the complexity of the model and improves the classification speed and accuracy. SVGPM \cite{ref18} used genetic programming to evolve SVM decision function, and enhances the recognition ability of minority class samples by adjusting the weight or structure of the loss function, so as to effectively solve the imbalanced classification problem. SSVM \cite{ref19} combined two smooth Hinge loss functions $\psi_G$ and $\psi_M$. These two loss functions are infinitely differentiable and uniformly converge to the traditional Hinge loss, so that second-order optimization algorithms such as TRON can be used to effectively improve the training efficiency. \cite{ref20} introduced a new smooth Hinge loss function, which combined Stochastic Gradient Descent (SGD) and Nesterov momentum to improve the convergence speed, solve the overfitting problem in high-dimensional data classification, and is suitable for imbalanced data scenarios such as insurance purchase prediction. In addition, deep learning methods, especially the combination of Convolutional Neural Network (CNN) and SVM, have achieved remarkable results in many fields, such as fingerprint recognition \cite{ref21}, biomedicina \cite{ref22} and so on. Via using the automatic feature extraction ability of deep learning and the powerful classification ability of SVM, the accuracy and robustness of the model can be significantly improved \cite{ref23,ref24}.

Another improvement is enhancing the robustness of SVM in handling noise and outliers by introducing different loss functions. Researchers have introduced several robust loss functions for Twin Support Vector Machines (TSVM), such as the symmetric LINEX loss function \cite{ref25}, the Welsch loss function \cite{ref26}, and the loss function induced by relevant entropy \cite{ref27}. While these loss functions contribute to improving the robustness of TSVM, they still have notable limitations. For instance, the symmetric LINEX loss function is affected by symmetry and unbounded growth, making it sensitive to extreme values, thus increasing sensitivity to outliers or noise. Akhtar et al. proposed the GL-TSVM \cite{ref28}, which effectively reduces the influence of outliers and enhances model robustness through the asymmetric, bounded, and smooth properties of the G-loss function. Additionally, other loss functions, such as the relevant entropy-induced loss \cite{ref27}, have been proposed to improve classification accuracy and generalization by increasing the penalty on misclassified samples. To reduce the training time of TSVM, Kumar and Gopal \cite{ref29} proposed the Least Squares TSVM (LSTSVM) algorithm. During the training of LSTSVM, the algorithm not only focuses on the support vector samples on the boundary but also considers some non-support vector samples, which enhances the model��s robustness and generalization capability. However, the squared error can significantly impact the error, leading to over-reliance on noise or outliers during the training process, thereby reducing generalization performance. To address the sensitivity of least squares loss to outliers, a non-convex least squares loss function \cite{ref30} was proposed, it helps TSVM achieve better classification results on more complex decision boundaries. However, this introduces challenges, particularly in the optimization process, where issues such as convergence to multiple local minima, slow convergence, and sensitivity to initial values may arise. Chen introduced a weighted version of LSTSVM (WLSTSVM) \cite{ref31}, which assigned different weights to error variables to eliminate the impact of noise and outliers. Arun Kumar et al. \cite{ref32} extended the LSTSVM model by incorporating prior knowledge in the form of multiple polytope sets, resulting in a new LSTSVM variant called KBLSTWSVM, which more effectively utilizes the information within the prior knowledge.

The Huber loss function \cite{ref33} provided a new solution for outlier-sensitive problems. It behaves similarly to the standard SVM loss function for small errors and suppresses the influence of outliers for large errors, thereby enhancing the model's stability in noisy data environments. Its hybrid form is akin to the $L_2$ norm penalty for small errors, and to the $L_1$ norm penalty for large errors. Generally, the $L_1$ norm penalty is more robust to outliers than the $L_2$ norm penalty. A robust $L_1$ norm non-parallel proximal support vector machine ($L_1$-NPSVM) was proposed, optimized using the gradient ascent algorithm to adapt to noisy environments \cite{ref34}. An $L_1$-norm distance-based twin support vector machine ($L_1$-TSVM) algorithm was introduced to enhance classification robustness \cite{ref35}. The $L_1$-norm dual projection support vector machine (TPSVM-$L_1$) was derived for image recognition applications \cite{ref36}. Additionally, the $L_1$-norm was incorporated into the twin support vector machine (CTSVM) to improve classification performance \cite{ref27}. The $L_{1-2}$ loss provides greater flexibility than both $L_1$ and $L_2$ by adjusting the parameter $p$. \cite{ref37} proposed sparse representation learning based on the $L_{1-2}$-norm, combining RRQR decomposition and SVM for optimal sensor placement in image classification, reducing computational complexity. However, the $L_{1-2}$ loss tends to have gradients close to zero for large errors, which may cause optimization stagnation, and the selection of the parameter p depends on empirical experience. Unlike the aforementioned loss functions, the Huber loss function divides the error into $L_2$ and $L_1$ regions using a parameter $\delta$. For small errors, the $L_2$ form is applied, ensuring a smooth and continuous gradient, while for large errors, it switches to the $L_1$ form to mitigate the impact of outliers. By adjusting $\delta$, the optimization process becomes more stable, improving the model's robustness and preventing overfitting.

The generalized adaptive mechanism can automatically adjust the parameters of the model during the training process to adapt to the changes between different data distributions and features. This adaptability can not make the model more flexible and robust, but deal with a variety of complex practical problems. At the same time, it lessens the need for manual parameter tuning while boosting the models' generalization ability and prediction accuracy \cite{ref38,ref39}. Hence, incorporating a generalized adaptive mechanism can significantly boost the model's robustness and adaptability, enhancing its performance and reliability in practical applications.

Drawing on the principles of the huber loss function, least squares loss function, and generalized adaptive mechanism, we propose a generalized adaptive huber loss function to enhance robustness to outliers and noise. In this paper, we further investigate the proposed twin support vector machine with generalized adaptive Huber loss function, named GAHTSVM. The proposed generalized adaptive loss function includes two parameters: an insensitive parameter $\epsilon (\epsilon \geq 0)$ and an adaptive robustness parameter $s (0\leq s\leq1)$. If the absolute value of the training error is below $\epsilon$, we follow the principle of the $\epsilon$-insensitive loss function and do not impose a penalty on this error. If the absolute value of the training error falls between $\epsilon$ and 1, we follow the principle of the Huber loss function and apply the $L_{2}$ norm penalty. If the absolute value of the training error exceeds 1, we apply the $L_{1}$ norm penalty to mitigate its negative impact on the final decision function by adjusting the robustness parameter $s$. Therefore, the adaptive robustness parameter $s$ determines the robustness, with the optimal value of $s$ being derived from the size and dimension of the dataset. Additionally, the $\epsilon$-insensitive loss function, the huber loss function, and the non-convex least squares loss function are all specific cases of the general non-convex quadratic $\epsilon$-insensitive loss function proposed. The proposed generalized adaptive huber loss function is non-convex, and the concave-convex procedure (CCCP) \cite{ref40} is employed to iteratively address a series of reconstructed convex optimization problems to tackle the original GAHTSVM problem. CCCP is essentially a particular form of DC (difference of convex) programming \cite{ref41,ref42}. Experimental results on 2 artficial datasets and 15 real-world datasets show that GAHTSVM outperforms TSVM, LSTSVM, HTSVM, CPin-GTBSVM, CNN-SVM, Wave-TSVM and SSVM by greatly reducing the effect of outliers or noise on decision functions, while demonstrating outstanding robustness. The principal contributions of this paper are:

(i) We present a generalized adaptive Huber loss function that has certain robustness and sparsity for classification problems.

(ii) A twin support vector machine classification method featuring a generalized adaptive Huber loss is introduced, offering enhanced robustness to noise through the adjustment of the adaptive robustness parameter.

(iii) The CCCP algorithm is used to solve this method and demonstrate its convergence.

(iv) Experimental results indicate that the proposed method demonstrates superior robustness compared to the other seven methods.

The remainder of this paper is organized as follows: Section 2 provides a brief overview of several classic and advanced loss functions and three related models: TSVM, LSTSVM, and HTSVM. Section 3 discusses the fundamental properties of the proposed generalized adaptive Huber loss and presents both the linear and nonlinear GAHTSVM. Section 4 details the experimental results on 2 artificial datasets and 15 real-world datasets. Finally, Section 5 offers the paper's conclusions.
\section{Related Work}\label{s2}
In this section, we will outline the background of the approach discussed in this paper and provide a brief introduction to three classification models: TSVM, LSTSVM, and HTSVM. To keep the discussion concise, we will focus solely on describing their models; additional details on the background work can be found in the respective references. In supervised learning tasks, the training set is typically provided as follows:
$D={(x_{i},y_{i})}_{i=1}^{m}=X_{1}\cup X_{2}$,
where \(x_i\in \mathbf{R}^{n}\) and \(y_i\in \{-1, 1\}\) denote the input and output respectively, $X_{1}=\{x_{i}|x_{i}\in \mathbf{R}^{n}, y_i=1, i=1, \ldots, m\}$, $X_{2}=\{x_{i}|x_{i}\in \mathbf{R}^{n}, y_i=-1, i=1, \ldots, m\}$. Let the positive and negative samples be denoted as $A=(x_{11}, \ldots, x_{1m_{1}})^{T}\in \mathbf{R}^{m_{1}\times n}$ and
$B=(x_{21}, \ldots, x_{2m_{2}})^{T}\in \mathbf{R}^{m_{2}\times n}$ respectively and $m= m_{1}+ m_{2}$.

In the following, we briefly review several representative loss functions and three widely adopted machine learning approaches: twin support vector machine (TSVM) ,least squares twin support vector machine (LSTSVM) and Functional iterative approaches for solving support vector classification problems based on generalized Huber loss (HTSVM).
\subsection{Loss Funcation}
Over an extended period, a multitude of researchers have been whole-heartedly dedicated to the exploration and development of novel loss functions. Their primary objective is to construct soft-margin support vector machine models. Subsequently, we will present a concise overview of the three loss functions pertinent to this paper in \autoref{loss}.

\begin{table}[htbp]
    \centering
    \caption{Characteristics of five loss functions with parameters $\delta = 0.5$, $\epsilon = 1$, and $a = 0.5$.}
    \label{loss}
    \scriptsize
    \begin{tabular}{m{2cm} m{4cm} m{4cm} m{4cm} m{2cm}}
        \toprule
        \textbf{Loss function} & \textbf{Mathematical formula} & \textbf{Advantage} & \textbf{Disadvantage} & \textbf{Figure}\\
        \midrule
        Huber loss &
        \(L_{\delta}(r) =
        \begin{cases}
            \delta(| r | - \frac{1}{2}\delta), & | r | > \delta \\
            \frac{1}{2}| r |^{2}, & | r | \leq \delta
        \end{cases}\) &
        \parbox{4cm}{The robustness of MSE \\and MAE is balanced to reduce \\ the influence of outliers.} &
        \parbox{4cm}{The sparsity is insufficient \\ and the computational \\ efficiency is \\ affected.} &
        \includegraphics[trim = 0mm 0mm 0mm 0mm, clip=true, width=2.5cm]{huber_fig.eps} \\
        $\epsilon$-insensitive loss &
        \(L_{\epsilon}(r) = (\max\{0,|r|-\epsilon\})^2\) &
        \parbox{4cm}{Small errors are ignored \\ and it is suitable \\ for noisy data.} &
        \parbox{4cm}{There are many support \\ vectors and the \\ computational cost is high.} &
        \includegraphics[trim = 0mm 0mm 0mm 0mm, clip=true, width=2.5cm]{epsilon_fig.eps} \\
        Non-convex least squares loss &
        \(L_{\delta}(r) =
        \begin{cases}
            \delta^2, & | r | > \delta \\
            r^{2}, & | r | \leq \delta
        \end{cases}\) &
        \parbox{4cm}{Flexible handling of \\ complex data.} &
        \parbox{4cm}{It is easy to fall into \\ local minima and \\ difficult to optimize.} &
        \includegraphics[trim = 0mm 0mm 0mm 0mm, clip=true, width=2.5cm]{non_convex_fig.eps} \\
        Smoothed Hinge loss &
        \(L_{\delta}(r) =~
        \begin{cases}
            0, & | r | \geq 1 \\
            1 - r - \frac{\delta}{2}, & | r | \leq 1 - \delta \\
            \frac{1}{2\delta}(1 - r)^2, & \text{otherwise}
        \end{cases}\) &
        \parbox{4cm}{Solve the non-differentiable \\ problem and improve \\ the stability of \\ optimization.} &
        \parbox{4cm}{Hyperparameter tuning \\ is complex.} &
        \includegraphics[trim = 0mm 0mm 0mm 0mm, clip=true, width=2.5cm]{smoothhinge_fig.eps} \\
        Wave loss &
        \parbox{4cm}{\(L_{\text{wave}}(r) =~\frac{1}{\delta}\left(1 - \frac{1}{1 + \delta r^{2}e^{ar}}\right),\\ \forall a \in \mathbb{R}\)} &
        \parbox{4cm}{Highly robust to outliers \\ and smoothness bounded.} &
        \parbox{4cm}{The parameter dependence \\ is strong and the cost of \\ parameter tuning is high.} &
        \includegraphics[trim = 0mm 0mm 0mm 0mm, clip=true, width=2.5cm]{wave_fig.eps} \\
        \bottomrule
    \end{tabular}
\end{table}

\subsection{TSVM}
In contrast to traditional SVM, TSVM \cite{ref7} aims to solve for a pair of non-parallel hyperplanes, one closer to the positive sample points and the other closer to the negative sample points, as described separately below:
$f_{1}(x)=\omega_{1}^{T}x+b_{1}=0$ and $f_{2}(x)=\omega_{2}^{T}x+b_{2}=0$, where $f_{1}(x)$ and $f_{2}(x)$ are the prediction functions for the positive and negative classes, respectively, while $\omega_{1}$ and $b_{1}$ denote the weight vector and bias for $f_{1}(x)$, similarly $\omega_{2}$ and $b_{2}$ denote the weight vector and bias for
$f_{2}(x)$. The formulation for linear TSVM classification is specified as follows:
\begin{equation}\label{1}
\begin{aligned}
&\min_{\omega_{1},b_{1}}~~\frac{1}{2}\|A\omega_{1}+e_{1}b_{1}\|_{2}^{2}+C_{1}e_{2}^{T}\xi_{1},\\
&s.t~~~-(B\omega_{1}+e_{2}b_{1})+\xi_{1}\geq e_{2},~~~\xi_{1}\geq0,
\end{aligned}
\end{equation}
and
\begin{equation}\label{2}
\begin{aligned}
&\min_{\omega_{2},b_{2}}~~\frac{1}{2}\|B\omega_{2}+e_{2}b_{2}\|_{2}^{2}+C_{2}e_{1}^{T}\xi_{2},\\
&s.t~~~~(A\omega_{2}+e_{1}b_{2})+\xi_{2}\geq e_{1},~~~\xi_{2}\geq0,~~~
\end{aligned}
\end{equation}
where $C_{1}\geq0$, $C_{2}\geq0$ denote the regularization parameters. Utilizing lagrange multipliers $\alpha \geq 0$, $\beta \geq 0$ along with the Karush-Kuhn-Tucker conditions, the dual formulations of the QPPs in \eqref{1} and \eqref{2} are given as follows:
\begin{equation}\label{3}
\begin{aligned}
&\min_{\alpha}~~\frac{1}{2}\alpha^{T}Q(P^{T}P)^{-1}Q^{T}\alpha- e_{2}^{T}\alpha,\\
&s.t~~~0\leq\alpha\leq C_{1}e_{2},
\end{aligned}
\end{equation}
and
\begin{equation}\label{4}
\begin{aligned}
&\min_{\beta}~~\frac{1}{2}\beta^{T}P(Q^{T}Q)^{-1}P^{T}\beta- e_{1}^{T}\beta,\\
&s.t~~~0\leq\beta\leq C_{2}e_{1},
\end{aligned}
\end{equation}
where $P=[A,e_{1}] $ and $Q=[B,e_{2}]$. Via solving \eqref{1} and \eqref{2}, the resulting proximal hyperplanes are as follows:
\begin{equation}\label{5}
\begin{aligned}
\begin{bmatrix}
	\omega_{1}&b_{1}
\end{bmatrix}^{T}=-(P^{T}P+\delta I)^{-1}Q^{T}\alpha,
\end{aligned}
\end{equation}
and
\begin{equation}\label{6}
\begin{aligned}
\begin{bmatrix}
\omega_{2}&b_{2}
\end{bmatrix}^{T}=(Q^{T}Q+\delta I)^{-1}P^{T}\beta,~~
\end{aligned}
\end{equation}
where $\delta I$ represents a regularization term, and $\delta$ is a positive constant.
\subsection{LSTSVM}
To decrease the training time for TSVM, Kumar and Gopal developed the LSTSVM \cite{ref29} algorithm. The primary benefit of LSTSVM compared to TSVM is its use of linear equations instead of QPPs, thereby reducing the model's computational complexity. Below are the primal problems associated with LSTSVM:
\begin{equation}\label{7}
\begin{aligned}
&\min_{\omega_{1},b_{1}}\frac{1}{2}\|A\omega_{1}+e_{1}b_{1}\|_{2}^{2}+\frac{C_{1}}{2}\xi_{1}^{T}\xi_{1},\\
&s.t~~-(B\omega_{1}+e_{2}b_{1})+\xi_{1}=e_{2},
\end{aligned}
\end{equation}
and
\begin{equation}\label{8}
\begin{aligned}
&\min_{\omega_{2},b_{2}}\frac{1}{2}\|B\omega_{2}+e_{2}b_{2}\|_{2}^{2}+\frac{C_{2}}{2}\xi_{2}^{T}\xi_{2},\\
&s.t~~~(A\omega_{2}+e_{1}b_{2})+\xi_{2}= e_{1},
\end{aligned}
\end{equation}
where $C_{1}\geq0$ and $C_{2}\geq0$ represent regularization parameters.

Insert the equality constraint into the objective function to derive:
\begin{equation}\label{9} \min_{\omega_{1},b_{1}}\frac{1}{2}\|A\omega_{1}+e_{1}b_{1}\|_{2}^{2}+\frac{1}{2}C_{1}\|e_{2}+B\omega_{1}+e_{2}b_{1}\|_{2}^{2},
\end{equation}
\begin{equation}\label{10}	\min_{\omega_{2},b_{2}}\frac{1}{2}\|B\omega_{2}+e_{2}b_{2}\|_{2}^{2}+\frac{1}{2}C_{2}\|e_{1}-A\omega_{1}-e_{1}b_{2}\|_{2}^{2}.
\end{equation}

Let $P=[A,e_{1}] $ and $Q=[B,e_{2}]$. We take the partial derivative of $\omega_{1}$ and $b_{1}$ , $\omega_{2}$ and $b_{2}$ to 0 in Eq.\eqref{7}and Eq.\eqref{8}, we can get:
\begin{equation}\label{11}
\begin{bmatrix}
	\omega_{1}&b_{1}
\end{bmatrix}^{T}=-(\frac{1}{C_{1}}P^{T}P+Q^{T}Q)^{-1}Q^{T}e_{2},
\end{equation}
and
\begin{equation}\label{12}
\begin{bmatrix}
\omega_{2}&b_{2}
\end{bmatrix}^{T}=(\frac{1}{C_{2}}Q^{T}Q+P^{T}P)^{-1}P^{T}e_{1}.~~~
\end{equation}

\subsection{HTSVM}
Due to the robustness and smooth transition characteristics of Huber loss function, it can usually produce models with better generalization ability. The primal problems of HTSVM \cite{ref33} are as follows:
\begin{equation}\label{13}
\begin{aligned}
&\min_{\omega_{1},b_{1}}\frac{1}{2}\|A\omega_{1}+e_{1}b_{1}\|_{2}^{2}+\frac{C_{1}}{2}(\|\xi_{2}\|^{2}-\|\xi_{2}-e_{1}\gamma\|^{2}-\|-\xi_{2}-e_{1}\gamma\|^{2}),\\
&s.t~~-(B\omega_{1}+e_{2}b_{1})+\xi_{2}=e_{2},
\end{aligned}
\end{equation}
and
\begin{equation}\label{14}
\begin{aligned}
&\min_{\omega_{2},b_{2}}\frac{1}{2}\|B\omega_{2}+e_{2}b_{2}\|_{2}^{2}+\frac{C_{2}}{2}(\|\xi_{1}\|^{2}-\|\xi_{1}-e_{2}\gamma\|^{2}-\|-\xi_{1}-e_{2}\gamma\|^{2}),\\
&s.t~~~~~~(A\omega_{2}+e_{1}b_{2})+\xi_{1}= e_{1},
\end{aligned}
\end{equation}
where parameter $\gamma$ serves as the balance point between quadratic loss and linear loss, $C_{1}\geq0$ and $C_{2}\geq0$ represent regularization parameters.

Plug the equality constraint into the objective function to get:
\begin{equation}\label{15}
\begin{aligned}
&\min_{\omega_{1},b_{1}}\frac{1}{2}\|A\omega_{1}+e_{1}b_{1}\|_{2}^{2}+\frac{C_{1}}{2}(\|B\omega_{1}+e_{2}b_{1}+e_{2}\|^{2}-\|B\omega_{1}+e_{2}b_{1}+e_{2}-e_{1}\gamma\|^{2}\\
&~~~~~~~~~-\|-(B\omega_{1}+e_{2}b_{1}+e_{2})-e_{1}\gamma\|^{2}),
\end{aligned}
\end{equation}
and
\begin{equation}\label{16}
\begin{aligned}
&\min_{\omega_{2},b_{2}}\frac{1}{2}\|B\omega_{2}+e_{2}b_{2}\|_{2}^{2}+\frac{C_{2}}{2}(\|A\omega_{2}+e_{1}b_{2}+e_{1}\|^{2}-\|A\omega_{2}+e_{1}b_{2}+e_{1}-e_{2}\gamma\|^{2}\\
&~~~~~~~~~-\|-(A\omega_{2}+e_{1}b_{2}+e_{1})-e_{2}\gamma\|^{2}).
\end{aligned}
\end{equation}

Assuming $P=[A~~~e_{1}] $ and $Q=[B~~~e_{2}]$, We take the partial derivative of $\omega_{1}$ and $b_{1}$ , $\omega_{2}$ and $b_{2}$ to 0 in Eq.\eqref{15} and Eq.\eqref{16}, we can get:
\begin{equation}\label{17}
\begin{bmatrix}
	\omega_{1}&b_{1}
\end{bmatrix}^{T}=-C_{1}(P^{T}P+C_{1}Q^{T}Q)^{-1}Qe_{2},
\end{equation}
and
\begin{equation}\label{18}
\begin{bmatrix}
	\omega_{2}&b_{2}
\end{bmatrix}^{T}=C_{2}(Q^{T}Q+C_{2}P^{T}P)^{-1}Pe_{1}.~~~
\end{equation}

\section{Generalized Adaptive Huber TSVM(GAHTSVM)}\label{s3}
In this section, we outline our proposed methods that use the robust huber loss function to address errors. At the same time, the linear and nonlinear versions of GAHTSVM are introduced in detail.

\subsection{\emph{Generalized Adaptive Huber Loss}}
Drawing from prior research, this study seeks to improve the robustness and sparsity of the losses discussed earlier. To achieve this goal, we introduce a new loss function that effectively combines both sparsity and robustness. The mathematical formulation of the proposed function is as follows:
\begin{eqnarray}\label{19}
L_{\epsilon }(r) =
\begin{cases}
    1 -\epsilon + s| r | - s, & | r | > 1, \\
    | r |^{2} - \epsilon | r |, & \epsilon \leq | r | \leq 1 ,\\
    0, & | r | < \epsilon,
\end{cases}
\end{eqnarray}
where $0 < \epsilon < 1$  is an insensitive parameter and $0 \leq s \leq 1$  is termed an adaptive parameter. The adaptive mechanism of this function automatically adjusts the value of $s$ based on changes in performance metrics. We will provide a detailed explanation of the adaptive mechanism of the parameter $s$ in Section 3.2.

Generalized adaptive Huber loss compared with the traditional $L_1$-norm, $L_2$-norm and Huber loss showed significant advantage, especially when dealing with noise and extreme samples. Via dynamically adjusting large error penalties and flexible sparsity processing, it makes up for the deficiencies of $L_1$-norm and $L_2$-norm in sparsity, robustness and optimization stability. Different from the traditional Huber loss with a fixed threshold, the generalized adaptive loss introduces an adjustable parameter \(s \) to dynamically adjust the penalty for large errors, which improves the adaptability and robustness of the model. In addition, by setting the zero loss interval \(|r| < \epsilon \), the penalty for small errors is reduced, the sparsity is enhanced, and the high computational cost is avoided. Although $L_{1-2}$-norm can effectively realize sparsification, it is computationally complex and may lead to overfitting on noisy data. In contrast, the generalized adaptive loss shows higher efficiency and robustness in the case of large-scale data and more noise through flexible parameter adjustment and tolerance interval \(\epsilon \), and its piecewise form further improves the computational efficiency.

Fig.\ref{Fig.GAH/obj} shows the performance of the proposed generalized adaptive Huber loss $L_{\epsilon}(r)$ for different values $s$ of the robustness parameter. The parameter $s$modulates the sensitivity of the model to outliers by controlling the severity of the penalty for large error intervals, that is, $|r| > 1$. In particular, When $s$ is large, the $s|r| - s$ term in the loss function increases for large errors with $|r| > 1$, making the penalty for large errors more severe. This encourages the model to focus more on reducing large errors, enhancing its sensitivity to noise, and applying stricter adjustments to samples with large errors. As a result, the model's accuracy improves by effectively minimizing the impact of large deviations. When
$s$ is small, the penalty for large errors decreases, allowing the model to tolerate larger deviations. This enables the model to better handle extreme samples and enhance its robustness, particularly when dealing with noisy data, where it can maintain strong performance.
\begin{figure}[H]
    \vspace{0.1cm}
    \begin{center}
            \centerline{\includegraphics[trim = 0mm 0mm 0mm 0mm, clip=true, width=9cm]{GAHLoss.eps}}
    \end{center}
\caption{The Generalized Adaptive Huber Loss Function with $\epsilon = 0.5$.}
\label{Fig.GAH/obj}
\vspace*{\fill}
\end{figure}

Obviously, it can be divided into three parts. When the absolute value of the error exceeds 1, the $L_{1}-$norm penalty dominates the effect of noise. This is similar to the huber loss, which also transitions to a linear form for large errors, thereby reducing the impact of noise on the loss. However, unlike the huber loss, the linear penalty of the proposed $L_{\epsilon}$ loss is more adaptable to large errors. By adjusting the parameter $s$, the $L_{\epsilon}$ loss can effectively detect noise, thereby mitigating their negative impact on the model. When the absolute value of the error lies between $\epsilon$ and 1, the $L_{\epsilon}$ loss adopts a smooth quadratic form. Within this range, the penalty for errors gradually increases, ensuring that small errors are not overly penalized, while larger errors receive progressively greater penalties. As the error $|r|$ varies within this interval, the function value depends on the value of $\epsilon$. A smaller $\epsilon$ can lead to a smaller function value in this range, which in turn assigns a lower weight to samples in this interval, promoting greater sparsity in the model. When the absolute value of the error is less than $\epsilon$, the $L_{\epsilon}$ loss is insensitive to small errors, forming a $\epsilon$-insensitive interval around the data. This means that samples corresponding to errors with an absolute value less than $\epsilon$ will be completely ignored, thus enhancing the sparsity of the model. In contrast, the Huber loss function still employs quadratic loss for small errors, which may lead to overfitting in some cases. The $L_{\epsilon}$ loss effectively avoids excessive penalty for small errors by assigning zero loss to errors smaller than $\epsilon$, thus improving the robustness of the model to noise.

Subsequently, we uncover the fundamental properties of the loss function.
\begin{theorem}
\label{theorem1}
$L_{\epsilon}(r)$ is symmetric and nonnegative.
\end{theorem}
\begin{proof}
From Eq.\eqref{19}, when $|r|>1$, $L_{\epsilon }(-r)=1 -\epsilon + s| -r | - s=1 -\epsilon + s| r | - s=L_{\epsilon }(r)$, and $L_{\epsilon }(r)=(1 -\epsilon) + s(| r | - 1)\geq 0$ since $\epsilon \leq 1$. When $\epsilon \leq | r | \leq 1$, $L_{\epsilon }(-r)=| -r |^{2} - \epsilon | -r |=| r |^{2} - \epsilon | r |=L_{\epsilon }(r)=| r |( | r |- \epsilon)\geq 0$. When $| r | < \epsilon$, $L_{\epsilon }(-r)=L_{\epsilon }(r)=0\geq 0$, i.e. $L_{\epsilon}(r)$ is symmetric and nonnegative.
\end{proof}
\begin{theorem}\label{theorem2}
(i) When $\epsilon \rightarrow 0$ and $s=1$, the proposed loss function is converted into a modified Huber-like loss function.
(ii) When $\epsilon \rightarrow 1$ and $s=1$, the proposed loss function is converted into the $\epsilon$-insensitive loss function.
\end{theorem}
\begin{proof}
(i) When $\epsilon \rightarrow 0$ and $s=1$, the original function is:
\begin{eqnarray}\label{20}
L_{\epsilon}(r) =
\begin{cases}
|r| & \text{if } |r| > 1, \\
|r|^2 & \text{if } |r| \leq 1.
\end{cases}
\end{eqnarray}
i.e. the modified Huber-like loss function.

(ii) When $\epsilon \rightarrow 1$ and $s=1$, the original function is:
\begin{eqnarray}\label{21}
L_{\epsilon}(r) =
\begin{cases}
|r| - 1 & \text{if } |r| > 1, \\
0 & \text{if } |r| \leq 1.
\end{cases}
\end{eqnarray}
This is the standard $\epsilon = 1$ insensitive loss.
\end{proof}
\begin{figure}[H]
\begin{minipage}{0.45\textwidth}
  \centerline{\includegraphics[trim = 0mm 0mm 0mm 0mm, clip=true, width=9cm]{4.eps}}
  \centerline{(a) Convex Loss Function}
\end{minipage}
\hfill
\begin{minipage}{0.45\textwidth}
  \centerline{\includegraphics[trim = 0mm 0mm 0mm 0mm, clip=true, width=9cm]{5.eps}}
  \centerline{(b) Concave Loss Function}
\end{minipage}
\caption{The Generalized Adaptive Huber Loss Function is Decomposed into A Concave Loss Function and A Convex Loss Function with $s=0.7$ and $\epsilon=0.5$.}
\label{Fig.2/obj}
\vspace*{\fill}
\end{figure}
\begin{theorem}
\label{theorem3}
The proposed loss function can be broken down into the sum of a concave function and a convex function.
\end{theorem}
\begin{proof}
$L_{\epsilon }(r)$ can be broken down into a concave part($L_{cave}(r)$) and a convex part($L_{vex}(r)$). ($L_{vex}(r)$) is as follows:
\begin{eqnarray}\label{22}
L_{vex}(r) =
\begin{cases}
    | r |^{2} - \epsilon | r | + s| r | - s, & | r | > 1, \\
    | r |^{2} - \epsilon | r |, & \epsilon \leq | r | \leq 1, \\
    0, & | r | < \epsilon,
\end{cases}
\end{eqnarray}
and ($L_{cav}(r)$) is defined as:
\begin{eqnarray}\label{23}
L_{cav}(r) =
\begin{cases}
    -| r |^{2} + \epsilon | r | + 1 - \epsilon, & | r | > 1, \\
    0, & | r | < 1.
\end{cases}
\end{eqnarray}
\end{proof}
\begin{remark}
Using the same value for parameter $s$, Fig.\ref{Fig.2/obj}(a) depicts the convex part $L_{vex}(r)$, while Fig.\ref{Fig.2/obj}(b) illustrates the concave part $L_{cav}(r)$.
\end{remark}

\subsection{Linear GAHTSVM}
Similarly to the TWSVM, the goal of GAHTSVM is to determine two prediction functions: $f_{1}(x)=\omega_{1}^{T}x+b_{1}=0$ and $f_{2}(x)=\omega_{2}^{T}x+b_{2}=0$.

Building on the loss function proposed in this paper, we consider the following linear form:
\begin{equation}\label{24} \min_{\omega_{1},b_{1}}\frac{1}{2}\|A\omega_{1}+e_{1}b_{1}\|_{2}^{2}
+C_{1}\sum_{i=1}^{m_{2}}L_{\epsilon}(1+(\omega_{1}^{T}x_{2i}+b_{1}))+\frac{1}{2}C_{3}(\|\omega_{1}\|_{2}^{2}+b_{1}^{2}),
\end{equation}
and
\begin{equation}\label{25} \min_{\omega_{2},b_{2}}\frac{1}{2}\|B\omega_{2}+e_{2}b_{2}\|_{2}^{2}
+C_{2}\sum_{i=1}^{m_{1}}L_{\epsilon}(1-(\omega_{2}^{T}x_{1i}+b_{2}))+\frac{1}{2}C_{4}(\|\omega_{2}\|_{2}^{2}+b_{2}^{2}).
\end{equation}

The GAHTWSVM problem under consideration can be separated into a convex part and a concave part as outlined below:
\begin{equation}\label{26}
\min_{\omega_{1},b_{1}}\underbrace{\frac{1}{2}\|A\omega_{1}+e_{1}b_{1}\|_{2}^{2}
+C_{1}\sum_{i=1}^{m_{2}}L_{vex}(1+(\omega_{1}^{T}x_{2i}+b_{1}))+\frac{1}{2}C_{3}(\|\omega_{1}\|_{2}^{2}+b_{1}^{2})}_{P_{vex}(\omega_{1},b_{1})}
+\underbrace{C_{1}\sum_{i=1}^{m_{2}}L_{cav}(1+(\omega_{1}^{T}x_{2i}+b_{1}))}_{P_{cav}(\omega_{1},b_{1})},
\end{equation}
and
\begin{equation}\label{27}
\min_{\omega_{2},b_{2}}\underbrace{\frac{1}{2}\|B\omega_{2}+e_{2}b_{2}\|_{2}^{2}
+C_{2}\sum_{i=1}^{m_{1}}L_{vex}(1-(\omega_{2}^{T}x_{1i}+b_{2}))+\frac{1}{2}C_{4}(\|\omega_{2}\|_{2}^{2}+b_{2}^{2})}_{P_{convex}(\omega_{2},b_{2})}
+\underbrace{C_{2}\sum_{i=1}^{m_{1}}L_{cav}(1-(\omega_{2}^{T}x_{1i}+b_{2}))}_{P_{cav}(\omega_{2},b_{2})}.
\end{equation}

We then apply CCCP to address the linear GTATSVM problem. The core approach of CCCP involves decomposing the original non-convex function into a convex part and a concave part, and then iteratively solving a sequence of convex optimization problems.
\begin{equation}\label{28}
\begin{bmatrix}
	\omega_{1}^{k+1}\\
	b_{1}^{k+1}
\end{bmatrix}=\arg\min\{P_{vex}(\omega_{1},b_{1})+
\begin{bmatrix}
(\nabla P_{cav}(\omega_{1}^k,b_{1}^k))^T
\end{bmatrix}\begin{bmatrix}
	\omega^{k}_{1}\\
	b^{k}_{1}
\end{bmatrix}\},
\end{equation}
and
\begin{equation}\label{29}
\begin{bmatrix}
	\omega_{2}^{k+1}\\
	b_{2}^{k+1}
\end{bmatrix}=\arg\min\{P_{vex}(\omega_{2},b_{2})+
\begin{bmatrix}
(\nabla P_{cav}(\omega_{2}^k,b_{2}^k))^T
\end{bmatrix}\begin{bmatrix}
	\omega^{k}_{2}\\
	b^{k}_{2}
\end{bmatrix}\}.
\end{equation}

Use $\theta_{i}^k$ and $\delta_{i}^k$ to represent the derivation result of $P_{cav}(\omega_{2},b_{2})$ and $P_{cav}(\omega_{2},b_{2})$.
\begin{eqnarray}\label{30}
\theta_{i}^{k} =C_{1}
\begin{cases}
    -2(1+(\omega_{1}^{T}x_{2i}+b_{1})) + \epsilon , & |1+(\omega_{1}^{T}x_{2i}+b_{1})| > 1 ,\\
    0, & 0 \leq |1+(\omega_{1}^{T}x_{2i}+b_{1})|  \leq 1,
\end{cases}
\end{eqnarray}
and
\begin{eqnarray}\label{31}
\delta_{i}^{k} =C_{2}
\begin{cases}
    -2(1-(\omega_{2}^{T}x_{1i}+b_{2})) + \epsilon , & |1-(\omega_{2}^{T}x_{1i}+b_{2})| > 1, \\
    0, & 0 \leq |1-(\omega_{2}^{T}x_{1i}+b_{2})|  \leq 1.
\end{cases}
\end{eqnarray}

By applying Eq.\eqref{30} and Eq.\eqref{31}, the subproblems defined by Eq.\eqref{26} and Eq.\eqref{27} can be respectively transformed into the following forms:
\begin{equation}\label{32} \min_{\omega_{1},b_{1}}\frac{1}{2}\|A\omega_{1}+e_{1}b_{1}\|_{2}^{2}
+C_{1}\sum_{i=1}^{m_{2}}L_{vex}(1+(\omega_{1}^{T}x_{2i}+b_{1}))+\sum_{i=1}^{m_{2}}\theta_{i}(1+(\omega_{1}^{T}x_{2i}+b_{1}))
+\frac{1}{2}C_{3}(\|\omega_{1}\|_{2}^{2}+b_{1}^{2}),
\end{equation}
and
\begin{equation}\label{33} \min_{\omega_{2},b_{2}}\frac{1}{2}\|B\omega_{2}+e_{2}b_{2}\|_{2}^{2}
+C_{2}\sum_{i=1}^{m_{1}}L_{vex}(1-(\omega_{2}^{T}x_{1i}+b_{2}))+\sum_{i=1}^{m_{1}}\delta_{i}(1-(\omega_{2}^{T}x_{1i}+b_{2}))
+\frac{1}{2}C_{4}(\|\omega_{2}\|_{2}^{2}+b_{2}^{2}).
\end{equation}

Note that $L_{vex}(r)=\max(0,(|r|^{2})-\epsilon |r|+\max(0,s|r|-s)$. By introducing the slack variables $\xi=(\xi_{1},\ldots,\xi_{m_{2}})$, $\eta=(\eta_{1},\ldots,\eta_{m_{1}})$ and let $\xi_{i}=1+(\omega_{1}^{T}x_{2i}+b_{1}),i=1,\ldots,m_{2}$, $\eta_{i}=1-(\omega_{2}^{T}x_{1i}+b_{2}),i=1,\ldots,m_{1}$ problems Eq.\eqref{26} and Eq.\eqref{27} become equivalent to:
\begin{equation}\label{34}
\begin{aligned}
&\min_{\omega_{1},b_{1},\xi}\frac{1}{2}\|A\omega_{1}+e_{1}b_{1}\|_{2}^{2}+C_{1}(\xi^{T}\xi+se_{2}^{T}\xi)+\sum_{i=1}^{m_{2}}\theta_{i}(1+(\omega_{1}^{T}x_{2i}+b_{1}))+\frac{1}{2}C_{3}(\|\omega_{1}
\|_{2}^{2}+b_{1}^{2})\\
&~~~\mbox{s.t.} ~~-(B\omega_{1}+e_{2}b_{1}) \geq \epsilon e_{2}-\xi,\xi \geq 0,\\
&~~~~~~~~~ -(B\omega_{1}+e_{2}b_{1}) \leq \epsilon e_{2}+\xi,
\end{aligned}
\end{equation}
and
\begin{equation}\label{35}
\begin{aligned}
&\min_{\omega_{2},b_{2},\eta}\frac{1}{2}\|B\omega_{2}+e_{2}b_{2}\|_{2}^{2}+C_{2}(\eta^{T}\eta+se_{1}^{T}\eta)+\sum_{i=1}^{m_{1}}\delta_{i}(1-(\omega_{2}^{T}x_{1i}+b_{2}))+\frac{1}{2}C_{4}(\|\omega_{2}
\|_{2}^{2}+b_{2}^{2})\\
&~~~\mbox{s.t.}~~  A\omega_{2}+e_{1}b_{2} \geq \epsilon e_{1}-\eta,\eta \geq 0,\\
&~~~~~~~~~ A\omega_{2}+e_{1}b_{2} \leq \epsilon e_{1}+\eta.
\end{aligned}
\end{equation}

To find the solution to problem \eqref{34}, we introduce the Lagrangian multipliers $\alpha_{1}$ and $\beta_{1}$. The lagrange function for this optimization problem \eqref{34} can then be expressed as:
\begin{eqnarray}\label{36}
% \nonumber to remove numbering (before each equation)
   L(\omega_{1},b_{1},\xi,\alpha_{1},\beta_{1})
   &=& \frac{1}{2}\|A\omega_{1}+e_{1}b_{1}\|_{2}^{2}+
   C_{1}(\xi^{T}\xi+se_{2}^{T}\xi)+\sum_{i=1}^{m_{2}}(\theta^{k})^T(\omega_{1}^{T}x_{2i}+b_{1}))+\frac{ C_{3}}{2}(\|\omega_{1}
\|_{2}^{2}+b_{1}^{2})\nonumber\\&+&\alpha_{1}^{T}(\epsilon e_{2}-\xi+B\omega_{1}+e_{2}b_{1})-\beta_{1}^{T}(B\omega_{1}+e_{2}b_{1}+\epsilon e_{2}+\xi).
\end{eqnarray}

The Karush-Kuhn-Tucker conditions need to be satisfied, and we can derive them as follows:
\begin{equation}\label{37}
\left\{\begin{aligned}
	&\frac{\partial L}{\partial \omega_{1}}=A^{T}(A\omega_{1}+e_{1}b_{1})+B^{T}\theta+C_{3} \omega_{1}+B^{T}\alpha_{1}-B^{T}\beta_{1}=0,~~~~&(i)&\\
	&\frac{\partial L}{\partial b_{1}}=e_{1}^{T}(A\omega_{1}+e_{1}b_{1})+e_{2}^{T} \theta+C_{3}b_{1}+e_{2}^{T}\alpha_{1}-e_{2}^{T}\beta_{1}=0, ~~~~&(ii)&\\
	&\frac{\partial L}{\partial \xi}=C_{1}\xi+\frac{C_{1}}{2}se_{2}-\alpha_{1}-\beta_{1}=0,~~~~&(iii)&\\
	&\alpha_{1}^{T}(B\omega_{1}+e_{2}b_{1}+\epsilon e_{2}-\xi)=0,\alpha_{1}\geq 0,~~~~&(iv)&\\
	&\beta_{1}^{T}[-(B\omega_{1}+e_{2}b_{1})-\epsilon e_{2}-\xi]=0, \beta_{1}\geq 0.~~~~&(v)&
\end{aligned} \right.
\end{equation}

Let $G=[A~e_{1}]$, $H=[B~e_{2}]$, $W_{1}=[\omega_{1}^{T}~b_{1}]^{T}$, $W_{2}=[\omega_{2}^{T}~b_{2}]^{T}$, then in combination (i) and (ii) in \eqref{37},we have~$G^{T}GW_{1}+C_{3}IW_{1}+H^{T}\alpha_{1}-H^{T}\beta_{1}+H^{T}\theta=0$, ~i.e.
\begin{equation}\label{38}
\begin{aligned}
W_{1}=-(G^{T}G+C_{3}I)^{-1}H^{T}(\alpha_{1}-\beta_{1}+\theta).~~~
\end{aligned}
\end{equation}

Similarly, we obtain
\begin{equation}\label{39}
\begin{aligned}
W_{2}=(H^{T}H+C_{4}I)^{-1}G^{T}(\alpha_{2}-\beta_{2}+\delta).~~~
\end{aligned}
\end{equation}

By (iii) in \eqref{37}, we have $\xi=\frac{1}{C_{1}}(\alpha_{1}+\beta_{1}-\frac{C_{1}}{2}se_{2})$. Thus \eqref{34} can be rewritten as
\begin{eqnarray}\label{40}
% \nonumber to remove numbering (before each equation)
   L(\omega_{1},b_{1},\xi,\alpha_{1},\beta_{1})
   &=& \frac{1}{2}(GW_{1})^{T}(GW_{1})+
   \frac{C_{1}}{2}(\xi^{T}\xi+se_{2}^{T}\xi)+\theta^{T}HW_{1}\nonumber\\&+&\frac{C_{3}}{2}W_{1}^{T}W_{1}+\alpha_{1}^{T}(\epsilon e_{2}-\xi+HW_{1})\nonumber+\beta_{1}^{T}[-HW_{1}-\epsilon e_{2}-\xi]\nonumber\\
   &=&-\frac{1}{2}(\alpha_{1}-\beta_{1}+\theta)^{T}H(G^{T}G+C_{3}I)^{-1}H^{T}(\alpha_{1}-\beta_{1}+\theta)
   +\epsilon e_{2}(\alpha_{1}-\beta_{1})\nonumber\\
   &-&\frac{1}{2C_{1}}(\alpha_{1}+\beta_{1}-\frac{C_{1}}{2}se_{2})^{T}(\alpha_{1}+\beta_{1}-\frac{C_{1}}{2}se_{2})\nonumber\\
   &=&-\frac{1}{2}[(\alpha_{1}-\beta_{1})^{T}H(G^{T}G+C_{3}I)^{-1}H^{T}(\alpha_{1}-\beta_{1})+\Theta_{1}^{T}H(G^{T}G+C_{3}I)^{-1}H^{T}(\alpha_{1}-\beta_{1})\nonumber\\
   &+&(\alpha_{1}-\beta_{1})^{T}H(G^{T}G+C_{3}I)^{-1}H^{T}\theta+\theta^{T}H(G^{T}G+C_{3}I)^{-1}H^{T}\theta]+\epsilon e_{2}(\alpha_{1}-\beta_{1})\nonumber\\
   &-&\frac{1}{2C_{1}}[(\alpha_{1}+\beta_{1})^{T}(\alpha_{1}+\beta_{1})-C_{1}se_{2}^{T}(\alpha_{1}+\beta_{1})+\frac{C_{1}^{2}}{4}s^{2}e_{2}^{T}e_{2}]
\end{eqnarray}

Remove irrelevant items of $\alpha_{1}$, $\beta_{1}$, and then obtain the dual problem \eqref{34}:
\begin{equation}\label{41}
\begin{aligned} &\min_{\alpha_{1},\beta_{1}}\frac{1}{2}[(\alpha_{1}-\beta_{1})^{T}H(G^{T}G+C_{3}I)^{-1}H^{T}(\alpha_{1}-\beta_{1})
   +(\alpha_{1}-\beta_{1})^{T}H(G^{T}G+C_{3}I)^{-1}H^{T}\theta\\
   &~~~~~~~+\theta^{T}H(G^{T}G+C_{3}I)^{-1}H^{T}(\alpha_{1}-\beta_{1})]
   -\epsilon e_{2}(\alpha_{1}-\beta_{1})+\frac{1}{2C_{1}}(\alpha_{1}+\beta_{1})^{T}(\alpha_{1}+\beta_{1})-se_{2}^{T}(\alpha_{1}+\beta_{1})\\
	&\mbox{s.t.}~~~~0\leq\alpha_{1},0\leq \beta_{1}.
\end{aligned}
\end{equation}

In a similar way, the dual problem of \eqref{35} is given by:
\begin{equation}\label{42}
\begin{aligned} &\min_{\alpha_{2},\beta_{2}}\frac{1}{2}[(\alpha_{2}-\beta_{2})^{T}G(H^{T}H+C_{4}I)^{-1}G^{T}(\alpha_{2}-\beta_{2})
   +(\alpha_{2}-\beta_{2})^{T}G(H^{T}H+C_{4}I)^{-1}G^{T}\delta\\
   &~~~~~~~+\delta^{T}G(H^{T}H+C_{4}I)^{-1}G^{T}(\alpha_{2}-\beta_{2})]
   -\epsilon e_{1}(\alpha_{2}-\beta_{2})+\frac{1}{2C_{2}}(\alpha_{2}+\beta_{2})^{T}(\alpha_{2}+\beta_{2})-se_{1}^{T}(\alpha_{2}+\beta_{2})\\
	&\mbox{s.t.}~~~~0\leq\alpha_{2},0\leq \beta_{2}.
\end{aligned}
\end{equation}

Let $\alpha=(\alpha_{1}-\beta_{1}),\beta=(\alpha_{2}-\beta_{2}),\Omega=H(G^{T}G+C_{3}I)^{-1}H^{T},\Lambda=G(H^{T}H+C_{4}I)^{-1}G^{T};
\mu=(\alpha_{1}+\beta_{1}),\nu=(\alpha_{2}+\beta_{2})$,then we can obtain
\begin{equation}\label{43}
\begin{aligned} &\min_{\alpha}\frac{1}{2}(\alpha^{T}\Omega\alpha+\theta^{T}\Omega\alpha+(\Omega\alpha)^{T}\theta)-\epsilon e_{1}\alpha+\frac{1}{2C_{1}}\mu^{T}\mu-se_{1}^{T}\mu\\
	&\mbox{s.t.}~~~~0\leq\alpha\leq C_{1}e_{2},
\end{aligned}
\end{equation}
and
\begin{equation}\label{44}
\begin{aligned} &\min_{\beta}\frac{1}{2}(\beta^{T}\Lambda\beta+\delta^{T}\Lambda\beta+(\Lambda\beta)^{T}\delta)-\epsilon e_{2}\beta+\frac{1}{2C_{2}}\nu^{T}\nu-se_{2}^{T}\nu\\
	&\mbox{s.t.}~~~~0\leq\beta\leq C_{2}e_{1}.
\end{aligned}
\end{equation}

For the adaptive parameter $s$, its value is automatically adjusted based on the sum of the proportions of outliers in both positive and negative samples. Specifically, for the outlier ratio $r_i$, if $r_i > a$, then $s=min(s * 1.1, 1.0)$; if $r_i < a$, then $s=max(s * 0.9, 0.0)$, where $a$ is a threshold, and the factors 1.1 and 0.9 are not fixed values and can be adjusted based on the specific context.

Using the iterative scheme described above, we derive the solution to the linear GAHTSVM problem with Algorithm1. Theorem4 verifies the convergence of the Algorithm1.
\begin{algorithm}[H]
\caption{CCCP for the Linear GAHTSVM}
\label{alg:framework}
\begin{algorithmic}
\Require
Training datasets \( A \in \mathbb{R}^{m \times n} \) and \( B \in \mathbb{R}^{m \times n} \); Parameters \( C_i \ (i = 1,2,3,4) \), \( \varepsilon \), tolerance \( tol \), maximal iteration number \( N \).
\Ensure
Decision function \( f(x) = \arg\min\left( \frac{|w_1^T x + b_1|}{\|w_1\|}, \frac{|w_2^T x + b_2|}{\|w_2\|} \right) \)
\State Initialize \( W_1 = \begin{bmatrix} w_1^T & b_1 \end{bmatrix}^T \), \( W_2 = \begin{bmatrix} w_2^T & b_2 \end{bmatrix}^T \); \( s = 0.1 \); iteration number \( k = 1 \).
\While{\( k \leq N \)}
    \State Compute \( r_1 = 1+(w_1^T x_{2i} + b_1) \), \( r_2 = 1 - (w_2^T x_{1i} + b_2) \)
    \State \( r = \frac{r_1 + r_2}{2} \)
    \If{\( r > a \)}
        \State \( s = \min(s \times 1.1, 1.0) \)
    \Else
        \State \( s = \max(s \times 0.9, 0.01) \)
    \EndIf
    \State Construct and solve the convex problems:
    \[
    W_1^{k+1} = \arg\min P_{vex}(w_1, b_1) + \left( \nabla P_{cav}(W_1^k) \right)^T W_1^k
    \]
    and
    \[
    W_2^{k+1} = \arg\min P_{vex}(w_2, b_2) + \left( \nabla P_{cav}(W_2^k) \right)^T W_2^k
    \]
    \State Obtain \( W_1^{k+1} \) and \( W_2^{k+1} \)
    \State Set \( k = k + 1 \)
    \If{\( \| W_1^{k} - W_1^{k-1} \| < tol \) \textbf{and} \( \| W_2^{k} - W_2^{k-1} \| < tol \)}
     $W_1^k$ and $W_2^k$ converges.
        \State \textbf{break}
    \EndIf
\EndWhile
\end{algorithmic}
\end{algorithm}
\subsection{Nonlinear GAHTSVM}
The GAHTSVM above constructs two non-parallel hyperplanes for linearly separable data. For nonlinear separable samples, the kernelized version of GAHTSVM is designed as follows. Before training the GAHTSVM model, the training set \(\mathcal{D} = \{(x_{i}, y_{i})\}_{i=1}^{m}\) must first be mapped into a higher-dimensional feature space using a kernel function \(K(\cdot, \cdot)\). It is important to note that the kernel function itself represents the nonlinear mapping. For any data point \(x_i\), its corresponding image in the feature space can be described as
\begin{equation}\label{45}
\Phi(x_i)={K(x_i, x_j)}_{j=1}^{m}\in \mathbf{R}^{m}.
\end{equation}

The results obtained for the linear GAHTSVM are now ready to be extended to the nonlinear case. To achieve this, the following two kernel-generated separating surfaces are considered:
\begin{equation}\label{46}
f_{1}(x)=\omega_{1}^{T}K(x,X^{T})+b_{1}=0
\end{equation}
and
\begin{equation}\label{47}
f_{2}(x)=\omega_{2}^{T}K(x,X^{T})+b_{2}=0
\end{equation}
where $X=[A^{T};B^{T}]^{T}$.

In order to obtain the two nonparallel planes \eqref{37} and \eqref{38}, we will consider a pair of the nonlinear optimization problems:
\begin{equation}\label{48}
\min_{\omega_{1},b_{1}}\frac{1}{2}\|K(A,X^{T})\omega_{1}+e_{1}b_{1}\|_{2}^{2}
+\sum_{i=1}^{m_{2}}L_{\epsilon}(1+\omega_{1}^{T}K(B,X^{T})+b_{1})
+\frac{1}{2}C_{3}(\|\omega_{1}\|_{2}^{2}+b_{1}^{2}),
\end{equation}
and
\begin{equation}\label{49}
\min_{\omega_{2},b_{2}}\frac{1}{2}\|K(B,X^{T})\omega_{2}+e_{2}b_{2}\|_{2}^{2}
+\sum_{i=1}^{m_{1}}L_{\epsilon}(1-\omega_{2}^{T}K(A,X^{T})-b_{2})
+\frac{1}{2}C_{4}(\|\omega_{2}\||_{2}^{2}+b_{2}^{2}),
\end{equation}
where $L_{\epsilon}$ is constructed in a manner identical to that of the linear version. An unseen sample $x\in \textbf{R}^{n}$ is classified into one of two classes according to the hyperplane it is closest to. For any input sample $x^{*}$, the classification decision function can be expressed as follows:
\begin{equation}\label{50}
f(x^{*})=argmin(\frac{|\omega_{1}^{T}K(x^{*},X^{T})+b_{1}|}{\sqrt{\omega_{1}^{T}K(X,X^{T})\omega_{1}}},~\frac{|\omega_{2}^{T}K(x^{*},X^{T})+b_{2}|}{\sqrt{\omega_{2}^{T}K(X,X^{T})\omega_{2}}}).
\end{equation}

Let $R=[K(A,X^{T})~e_{1}]$, $S=[K(B,X^{T})~e_{2}]$, $\widetilde{\Omega}=S(R^{T}R+C_{3}I)S^{T}$, $\widetilde{\Lambda}=R(S^{T}S+C_{3}I)R^{T}$, $\widetilde{\mu}=(\widetilde{\alpha}_{1}+\widetilde{\beta}_{1}),\widetilde{\nu}=(\widetilde{\alpha}_{2}+\widetilde{\beta}_{2})$, the QPPs for the $k-$th nonlinear GAHTSVM within the CCCP algorithm can be expressed as:
\begin{equation}\label{51}
\begin{aligned} &\min_{\widetilde{\alpha}}\frac{1}{2}(\widetilde{\alpha}^{T}\widetilde{\Omega}\widetilde{\alpha}+\widetilde{\theta}^{T}\widetilde{\Omega}\widetilde{\alpha}+(\widetilde{\Omega}\widetilde{\alpha})^{T}\widetilde{\theta})-\epsilon e_{1}\widetilde{\alpha}+\frac{1}{2C_{1}}\widetilde{\mu}^{T}\widetilde{\mu}-se_{1}^{T}\widetilde{\mu}\\
	&\mbox{s.t.}~~~~0\leq\widetilde{\alpha}\leq C_{1}e_{2},
\end{aligned}
\end{equation}
and
\begin{equation}\label{52}
\begin{aligned} &\min_{\widetilde{\beta}}\frac{1}{2}(\widetilde{\beta}^{T}\widetilde{\Lambda}\widetilde{\beta}+\widetilde{\delta}^{T}\widetilde{\Lambda}\widetilde{\beta}+(\widetilde{\Lambda}\widetilde{\beta})^{T}\widetilde{\delta})-\epsilon e_{2}\widetilde{\beta}+\frac{1}{2C_{2}}\widetilde{\nu}^{T}\widetilde{\nu}-se_{2}^{T}\widetilde{\nu}\\
	&\mbox{s.t.}~~~~0\leq\widetilde{\beta}\leq C_{2}e_{1},
\end{aligned}
\end{equation}
where
\begin{eqnarray}\label{53}
\widetilde{\theta}_{i}^{k} =C_{1}
\begin{cases}
    -2(1+\omega_{1}^{T}K(B,X^{T})+b_{1})) + \epsilon , & |1+\omega_{1}^{T}K(B,X^{T})+b_{1})| > 1 ,\\
    0, & 0 \leq |1+(\omega_{1}^{T}K(B,X^{T})+b_{1})|  \leq 1,
\end{cases}
\end{eqnarray}
and
\begin{eqnarray}\label{54}
\widetilde{\delta}_{i}^{k} =C_{1}
\begin{cases}
    -2(1-(\omega_{2}^{T}K(A,X^{T})+b_{2})) + \epsilon , & |1-(\omega_{2}^{T}K(A,X^{T})+b_{2})| > 1, \\
    0, & 0 \leq |1-(\omega_{2}^{T}K(A,X^{T})+b_{2})|  \leq 1.
\end{cases}
\end{eqnarray}
\subsection{Convergence Analysis}
In order to prove that GAHTSVM achieves convergence by the CCCP algorithm, we first present Theorem4 and prove the convergence of Algorithm1.
\begin{theorem}
\label{theorem4}
Algorithm 1 ensures a monotonic decrease in the objective of problems \eqref{34} and \eqref{35} with each iteration.
\end{theorem}
\begin{proof}
Let $W=[w^{T}~~~b]^{T}$, $W^{k+1}$ denote the optimal solution to problems \eqref{30} and \eqref{31} at the $k$-th iteration of Algorithm 1. The objective function of GAHTSVM can be decomposed as follows:
\begin{equation}\label{55}
\min_{W} P(W) = P_{vex}(W) - P_{cav}(W).
\end{equation}
where \(P_{vex}(W)\) is a convex function, \(P_ {cav} (W)\) is a concave function. By the CCCP algorithm, the concave function \(P_{cav}(W)\) is linearized in each iteration and transformed into a convex subproblem:
\begin{equation}\label{56}
W^{(k+1)}=\arg\min_{W}[P_{vex}(W)-\bigtriangledown P_{cav}(W^{k})(W-W^{k})].
\end{equation}

The convex nature of $P_{vex}(W)$ means that
\begin{equation}\label{57}
P_{vex}(W^{k+1})\geq P_{vex}(W^{k})+(W^{k+1}-W^{k})\cdot \bigtriangledown P_{vex}(W^{k}).
\end{equation}

The concave nature of $P_{concave}(W)$ means that
\begin{equation}\label{58}
P_{cav}(W^{k+1})\leq P_{cav}(W^{k})+(W^{k+1}-W^{k})\cdot \bigtriangledown P_{cav}(W^{k}).
\end{equation}

Using the discrete iteration $\bigtriangledown P_{vex}(W^{k+1})=-\bigtriangledown P_{cav}(W^{k})$, we find that
\begin{equation}\label{59}
P_{vex}(W^{k+1})+P_{cav}(W^{k+1})\leq P_{vex}(W^{k})+P_{cav}(W^{k}).
\end{equation}
i.e. \(P(w^{(k+1)}) \leq P(w^{(k)})\), the objective function is monotonically decreasing.
\end{proof}

In terms of each iterations, algorithm 1 firstly updates the model parameter $W_1$ by optimizing the convex part $P_{vex} (W_1)$, then uses $\bigtriangledown P_{cav} (W_1^k)$to calculate the gradient. To ensure updates in the direction of $\bigtriangledown P_{cav} (W_1)$. Since the lower bound of problems \eqref{34} and \eqref{35} are 0, algorithm1 converges. Thus, the target values for problems \eqref{34} and \eqref{35} are reduced in each iteration until the algorithm reaches convergence.
\subsection{Noise Insensitivity and Sparsity}
In this section, we provide a theoretical explanation of how the $L_{\epsilon}$ loss function enhances robustness to noise. For simplicity, we use the linear case as an example, though the same analysis applies to the nonlinear case. The subgradient of the $L_{\epsilon}$ loss function is as follows:
\begin{eqnarray}\label{60}
g_{\epsilon}(x) =
\begin{cases}
    -s, & x < -1, \\
    \{-s,2x + \epsilon\}, & x = -1, \\
    2x + \epsilon, & -1 < x < -\epsilon, \\
    \{2x + \epsilon,0\}, & x = -\epsilon, \\
    0, & -\epsilon < x < \epsilon, \\
    \{0,2x - \epsilon\}, & x = \epsilon, \\
    2x - \epsilon, & \epsilon < x < 1, \\
    \{2x - \epsilon,s\}, & x = 1, \\
    s, & x > 1.
\end{cases}
\end{eqnarray}

In addition, we partition the data points in the $i-$th class into nine subsets:
\begin{eqnarray*}
& T_{1}&=\{m:1-|w_{i}^{T}x_{m}+b_{i}| < -1\}, \\
& T_{2}&=\{m:1-|w_{i}^{T}x_{m}+b_{i}| = -1\}, \\
& T_{3}&=\{m:-1 < 1-|w_{i}^{T}x_{m}+b_{i}| < -\epsilon\}, \\
& T_{4}&=\{m:1-|w_{i}^{T}x_{m}+b_{i}| = -\epsilon\}, \\
& T_{5}&=\{m:-\epsilon < 1-|w_{i}^{T}x_{m}+b_{i}| < \epsilon\}, \\
& T_{6}&=\{m:1-|w_{i}^{T}x_{m}+b_{i}|= \epsilon\}, \\
& T_{7}&=\{m:\epsilon < 1-|w_{i}^{T}x_{m}+b_{i}| < 1\}, \\
& T_{8}&=\{m:1-|w_{i}^{T}x_{m}+b_{i}| = 1\}, \\
& T_{9}&=\{m:1-|w_{i}^{T}x_{m}+b_{i}| > 1\}.
\end{eqnarray*}

It is easy to find that larger \(s\) can effectively suppress samples with large errors, reduce their interference to model training, increase the number of support vectors, especially the samples near the decision boundary, so as to improve the robustness of the model and help control the sparsity of the model. Increasing \(\epsilon \) can reduce the influence of samples with smaller errors, make more samples enter the region with zero gradient, reduce the influence of these samples, and further improve the robustness and sparsity.
\begin{remark}
For samples belonging to \(T_1, T_9 \), which are far from the decision boundary, the gradient of the loss function is constant \(\pm s \), indicating that the effect of these samples is directly related to the size \(s \). Larger \(s \) can effectively reduce the impact of these samples on model updating, thereby enhancing robustness. At the same time, it will make the model carry out a larger weight update to the samples far away from the boundary, thus promoting sparsity.
\end{remark}
\begin{remark}
For the points in the set \(T_5 \), the gradient of the loss function is zero, indicating that these samples no longer affect the model. When \(\epsilon \) increases, more samples enter the region, and the model will be more dependent on the few key samples close to the decision boundary, reducing the influence of irrelevant samples on training, and thus improving the sparsity and robustness of the model.
\end{remark}
\begin{remark}
For the samples in the sets \(T_3 \) and \(T_7 \), their errors are large but do not exceed the critical value, and the influence is more significant. Reducing \(s \) can effectively reduce the interference of these samples and further improve the sparsity of the model.
\end{remark}
\begin{remark}
For the samples in the sets \(T_2, T_4, T_6, T_8 \), they lie in the critical region of the loss function and are crucial for the selection of support vectors. Via adjusting \(s \) and \(\epsilon \), we can fine-tune which samples become support vectors and thus affect the sparsity of the model.
\end{remark}

Consequently, via reasonable adjustment of \(s \) and \(\epsilon \), we can ensure the robustness of the model, further improve the sparsity, and make the model more efficient and accurate.
\subsection{Computational Convexity}
Eight methods: TSVM, LSTSVM, HTSVM, CNNSVM, CPin-TBSVM, Wave-TSVM, SSVM and GAHTSVM have similarities and differences, and we compare GAHTSVM with the other seven algorithms. For ease of comparison, $m$ is set as the number of samples, and $m_1$ and $m_2$ are respectively the number of positive and negative training samples, that is, $m = m_1 + m_2$. $M$ represents the number of iterations, $n$ is the feature dimension. The proposed GAHTSVM is sparse and insensitive to noise. This problem is not a convex optimization problem, but can be solved by CCCP method. Since these parts are executed in each iteration cycle, the overall computational complexity is roughly $O(M \times ((n+1)^3 + m \times n^2))$. The number of iterations $M$ depends on the parameters $C_{i} \ (i = 1, 2, 3, 4), \epsilon$, so it is difficult to predict. However, experiments show that the number of iterations is usually less than 100.

\textbf{Comparison with TWSVM.} TWSVM decomposes the problem into two subproblems, each with constraints from data points of the other class. Assuming that the number of samples in the two classes is roughly balanced (\(m_1 \approx m_2 \approx m/2\)), the number of variables in each subproblem is \(n+1\) and the number of constraints is \(m/2\). Therefore, the complexity of each subproblem is \(O((m/2)^3)\). The overall complexity is \(2 \times O((m/2)^3) = O(m^3/4)\).

\textbf{Comparison with LSTSVM.} LSTSVM transforms each subproblem into a system of linear equations, and the complexity of matrix inversion is \(O((n+1)^3)\), and the total complexity is \(O(2 \times (n+1)^3)\). When \(n \ll m\) (such as high-dimensional sparse data in text classification), the complexity of LSTSVM is much lower than TSVM.

\textbf{Comparison with HTSVM.} Each iteration requires solving two equations of linear equations, Involved matrix \((\frac {E ^ tE} {\lambda_1} + F ^ tF + \varepsilon I) ^ {1} \) and \((\frac {F ^ tF} {\lambda_2} + E ^ tE + \varepsilon I)^{-1}\), the complexity is \(O(n+1)^3)\). Optimized by the Sherman-Morrison-Woodbury (SMW) formula, it can be decomposed into the inverse of a smaller matrix, and the complexity is reduced to \(O(m_1^3 + m_2^3)\).

\textbf{Comparison with CPin-TBSVM.} CPin-TBSVM is based on the research of Pin-TBSVM and the upper limit pinball loss function. It is designed to solve the non-convex QPP problem with CCCP algorithm. This method has successfully solved a series of QPP problems. During the solution process, the computational complexity of CPin-TBSVM is $O(M \times (m/2)^3)$.

\textbf{Comparison with CNN-SVM.} The time complexity of CNN-SVM is mainly determined by the training process of CNN and SVM. The time complexity of CNN is roughly $O(n_1 \times n\times d)$, where $n_1$ is the number of training samples, and $d$ is the number of network parameters. SVM training complexity is usually $O(m^2)$. The overall time complexity is $O(n_1 \times n \times d + m^2)$, which depends on the feature extraction of CNN and the classification process of SVM.

\textbf{Comparison with Wave-TSVM.} The computational complexity of Wave-TSVM mainly comes from the calculation of matrix inversion. In the linear case, the algorithm needs to solve the inverse of $(n + 1) \times (n + 1)$ matrix, so the time complexity is $O((n + 1)^3)$. For the nonlinear case, the algorithm needs to compute the inverse of two matrices: one is of size $m_1\times m_1$, the other is of size $m_2 \times m_2$, and the computational complexity is $O(m_1^3)$ and $O(m_2^3)$, respectively. Therefore, the time complexity of nonlinear Wave-TSVM is $O(M(m_1^3 +m_2^3))$.

\textbf{Comparison with SSVM.} The computational complexity of SSVM mainly depends on the optimization algorithm employed. The computational complexity of SSVM combined with TRON is \(O(m \cdot n + k \cdot M)\), and quadratic convergence is achieved through second-order optimization. SSVM has both efficiency and accuracy on high-dimensional sparse data, and is suitable for large-scale machine learning tasks.
\section{Numerical Experiments}\label{s4}
In this section, we will present the relevant experiments and analyses of results pertaining to GAHTSVM, with a focus on elucidating its various characteristics through data analysis. We will evaluate the performance and effectiveness of GAHTSVM through a series of experiments utilizing both real-world and synthetic datasets, comparing its outcomes with those of traditional support vector machine algorithms to underscore GAHTSVM's advantages in classification tasks. Finally, we will conclude this section by summarizing the experimental findings and outlining potential avenues for future research and enhancements related to GAHTSVM. The implementations of these methods were carried out in MATLAB R2024a on a PC with a 64-bit Windows XP OS and a 3.10 GHz processor.

\subsection{Experimental Setting}
In the experimental setup, we conducted a comparative analysis of the real-world dataset and an artificial dataset, employing 10-fold cross-validation to partition the data into training and testing sets. This approach was implemented to ensure both the stability and reliability of our results.

To assess the classification performance of our proposed GAHTSVM, we systematically compared it with several advanced robust methods, including SSVM, Wave-TSVM, CNN-SVM, CPin-GTBSVM, LSTSVM, HTSVM, and TSVM. These algorithms have demonstrated strong performance in classification tasks and exhibit significant potential for practical application, as illustrated in the accompanying.
\begin{table}[htbp]
\caption{Comparison of Algorithms and Their Related Descriptions}
\scriptsize
\begin{center}
\begin{tabular}{l l}
\toprule
\textbf{Algorithm} & \textbf{Description} \\
\midrule
CPin-GTBSVM & Twin Bounded Support Vector Machine with Capped Pinball Loss \cite{ref9} / 2024. \\
Wave-TSVM&Advancing supervised learning with the wave loss function: A robust and smooth approach.\cite{ref15}/2024.\\
SSVM&Learning with smooth Hinge losses.\cite{ref19}/2021.\\
HTSVM & Functional iterative approaches for solving support vector classification problems\\
 &based on generalized Huber loss \cite{ref33} / 2020. \\
CNN-SVM & A novel hybrid CNN�CSVM classifier for recognizing handwritten digits \cite{ref23} / 2012. \\
LSTSVM & Least squares twin support vector machines for pattern classification \cite{ref29} / 2009. \\
TSVM & Twin support vector machines for pattern classification \cite{ref7} / 2007. \\
\bottomrule
\end{tabular}
\end{center}
\label{tab:algorithm}
\end{table}

In this experiment, we primarily focused on analyzing the nonlinear behavior exhibited by each method and selected the Gaussian radial basis function (RBF) $\mathbb{K}(x,x')=\exp(-\frac{\|x-x'\|^{2}}{2\delta^{2}})$ for all algorithms involved.
In order to evaluate the classification performance of six algorithms, the accuracy metric (ACC) was employed. The formula for calculating this metric is as follows:
\begin{eqnarray}\label{62}
ACC=\frac{TP+TN}{TP+FN+TN+FP},
\end{eqnarray}
where TP and TN indicate true positives and true negatives, and FP and FN signify false positives and false negatives, respectively. Further, we conducted a comparative analysis of the training time between the proposed GAHTSVM and the baseline models. The reported times only include the time taken for training the models with the optimized hyperparameters.
\subsection{Artificial Datasets}
In the artificial datasets (a) and (b), each contains 50 positive and 50 negative samples, which are denoted by $+$, $*$ and $\square$, respectively, as shown in Fig.\ref{Fig.4/obj}. Since outliers may have an impact on the classification performance, the stability of the algorithm also becomes one of the measures. Therefore, we introduce four outliers in the artificial dataset to evaluate the robustness of the algorithm, where two outliers belong to the class $+1$ and the other two belong to the class $-1$.

Outliers often have a certain impact on classification performance, which is also a measure of algorithm stability. Therefore, we introduce outliers in the artificial data sets (a) and (b), respectively, as shown intuitively in Fig.\ref{Fig.4/obj}. In order to further verify the robustness of the generalized adaptive loss function, we show the classification accuracy of this algorithm on these two data sets in Fig.\ref{Fig.art/obj}, and compare it with other seven algorithms.

As can be seen from Fig.\ref{Fig.art/obj}, the proposed GAHTSVM algorithm shows higher accuracy in the face of outliers.On the artificial dataset (a) and (b), the classification accuracy of GAHTSVM reaches 68.98\% and 93.75\% respectively, which is significantly better than the other seven algorithms, showing its excellent performance in dealing with outliers. GAHTSVM can effectively deal with the interference of outliers, and has stronger robustness and superior classification ability.
\begin{figure}[H]
\begin{minipage}{0.45\textwidth}
  \centerline{\includegraphics[trim = 0mm 0mm 0mm 0mm, clip=true, width=8cm]{xor.eps}}
  \centerline{(a) Artficial Dataset1.}
\end{minipage}
\hfill
\begin{minipage}{0.45\textwidth}
  \centerline{\includegraphics[trim = 0mm 0mm 0mm 0mm, clip=true, width=8cm]{two_moon.eps}}
  \centerline{(b) Artficial Dataset2.}
\end{minipage}
\caption{Artificiatial Datasets with Outliers.}
\label{Fig.4/obj}
\end{figure}
\begin{figure}[H]
    \centering
    \includegraphics[trim = 0mm 0mm 0mm 0mm, clip=true, width=11cm]{art_ACC.eps}
    \caption{Accuracy of Two Artificial Datasets with Outliers.}
    \label{Fig.art/obj}
\end{figure}
In conclusion, the GAHTSVM algorithm can effectively overcome the adverse effects caused by outliers in the experiment and help the model reduce the interference of outliers on the classification results. In particular, when the adaptive parameter $s$ is relatively large, the penalty for outliers can be increased, thus enhancing the adaptability of the model to abnormal data.
\subsection{Real-world Datasets}
In order to verify the effectiveness of the proposed GAHTSVM, we will present a comparison experiment with five other algorithms based on 15 real-world datasets.
These 15 datasets are all sourced from the UCI Machine Learning Repository and represent classical benchmark data across a wide range of fields. Among the above 15 datasets, there are 5 datasets each with low, medium, and high noise levels. Among them, the high noise level datasets include a8a, Spam, Detecting, Hepatitis, Cancer. The medium noise level dataset covers Australian, German, Ionosphere, Spectf, WDBC. The low noise level dataset contains Bank, Mushroom, Vote, Wholesale, QSAR. Among them, the Australian (1987) and German (1994) datasets are credit evaluation datasets for Australia and Germany, with sample sizes of 690 and 1,000, respectively. The Bank (2008) dataset consists of 250 customer records from a Portuguese bank's telemarketing campaign. The medical domain includes the Wisconsin breast cancer fine-needle aspiration dataset from 1989 (699 cases) and the WDBC breast cancer image feature dataset from 1995 (569 cases). The Detecting (2013) dataset, from remote sensing, contains 3,292 hyperspectral image samples with 1,084 dimensional features. The QSAR bioinformatics dataset (2010) includes 1,055 chemical molecules for activity prediction. In the text classification domain, there is the 1999 Spam dataset (4,131 entries) and the preprocessed a8a text data (originally from the 1990s). Additionally, we include the Mushroom fungus classification data from the 1980s (8,124 entries), the US Congressional voting records from 2004 (435 entries), the Wholesale customer segmentation dataset from 2005 (440 entries), the Ionospheric radar dataset (306 entries), the Hepatitis diagnosis dataset (155 entries), and the Spectral feature selection dataset (267 entries). All datasets have been standardized by the UCI repository, and they span a time range from the 1980s to 2015, covering a variety of research scenarios such as classification, feature selection, and remote sensing detection. For more detailed information about the reported datasets, please visit the official UCI website: http://mlr.cs.umass. edu/ml/index.html. Each dataset has different characteristics and categories, covering multiple domains and question types, detailed in \autoref{tab:UCI}, including dataset name, sample size, and number of features. In addition, we will further analyze and discuss the experimental results, explore the advantages and limitations of GAHTSVM algorithm, and provide a reasonable explanation of the experimental results.
\begin{table}[htbp]
\caption{Characteristics of Real-World Datasets}
\label{tab:UCI}
\scriptsize
\begin{center}
\begin{tabular}{lcc lcc}
\toprule
\textbf{Datasets} & \textbf{Samples} & \textbf{Attributes} & \textbf{Datasets} & \textbf{Samples} & \textbf{Attributes} \\
\midrule
a8a & 9865 & 123 & Australian & 690 & 14 \\
Bank & 250 & 6 & Cancer & 699 & 9 \\
Detecting & 3292 & 1084 & German & 1000 & 24 \\
Hepatitis & 155 & 19 & Ionosphere & 306 & 34 \\
Mushroom & 8124 & 112 & QSAR & 1055 & 41 \\
Spam & 4131 & 59 & Spectf & 267 & 44 \\
Vote & 435 & 16 & WDBC & 569 & 30 \\
Wholesale & 440 & 7 & & & \\
\bottomrule
\end{tabular}
\end{center}
\end{table}

To obtain more objective experimental results, we normalized the datasets used in the experiment to ensure that the data remained within the interval $[0, 1]$, thereby minimizing the differences between various sample features. We conducted 10-fold cross-validation across all datasets, a method that mitigates the randomness associated with single test results by employing multiple training and testing dataset splits, thus enhancing the model's robustness. Given that noise is a key criterion for evaluating an algorithm's robustness, we will examine data points that include noise. If the classification accuracy varies smoothly as noise increases, it indicates that the algorithm exhibits strong resistance to noise. In addition, we added zero-mean Gaussian noise to simulate the feature noise, and the ratio of its variances was set to 0.1 and 0.3 (r = 0.1, 0.3).
\subsection{Parameter Analysis}
Besides the choice of data sets and processing, parameter selection and the key factors influencing the model performance. The penalty parameter $C_{i}$ plays a role in balancing training error and model complexity in the model. At the beginning of the experiment, we set the $C_ {1} = C_ {2} $, $C_ {3} = C_ {4} $, and set the scope to $\{10^{i} \mid i=-5,-4,-3,-2,-1,0,1,2,3,4,5\}$, covered from lesser penalties to the wide range of large penalties. In order to more accurately investigate the impact of $C_{i}$ on model performance, we plotted 3D bar charts of $C_{i}$ related to ACC for six representative datasets (which include two high-noise datasets: Detecting and Hepatitis, three medium-noise datasets: Australian, Ionosphere, and WDBC, and one low-noise dataset: QSAR), as shown in the Fig.\ref{Fig.9/obj}.

From Fig.\ref{Fig.9/obj}, it can be observed that different datasets exhibit varying sensitivities to $C_{i}$. In the Australian dataset, when $C_{i}$ takes a lower value, such as close to $10^{-3}$, the ACC value is relatively low, indicating that the model, with insufficient regularization, fails to fully learn the data features, resulting in underfitting. As the value of $C_{i}$ increases, the ACC value gradually rises. However, once $C_{i}$ exceeds $10^{2}$, the growth of the ACC value starts to flatten and may even decrease in some regions. This could be due to the excessive regularization, causing the model to focus too much on the training data, leading to overfitting.

The same trend is observed in other datasets as well. Via synthesizing the experimental results from multiple datasets, we find that in most cases, when \( C_{i} \) takes values in \( \{10^{i} \mid i=-1, 0, 1, 2\} \), the ACC value is higher and more stable. Therefore, in subsequent experiments, to more effectively search for the optimal parameter, we narrow the range of values for \( C_{i} \) to \( \{10^{i} \mid i= -1, 0, 1, 2\} \).

The insensitive parameter $\epsilon$ defines the region where the model is insensitive to errors, meaning that errors within the range of $\epsilon$ are not counted in the loss function. In preliminary experiments, we set the range of $\epsilon$ to be from 0.1 to 1. To determine a more suitable range, we also plotted 2D bar charts of $\epsilon$ related to ACC for the six datasets mentioned above. From Fig.\ref{Fig.8/obj}, it can be observed that the impact of $\epsilon$ on ACC exhibits a certain pattern across different datasets. When the value of $\epsilon$ is small, the ACC values of the model are relatively low on most datasets, possibly because the model is too sensitive to errors, leading to overfitting to the noise in the training data. As the value of $\epsilon$ increases, the ACC value rises within a certain range. However, when $\epsilon$ exceeds 0.7, the ACC value begins to decline, indicating that at this point, the model's tolerance for errors is too high, causing it to fail to accurately capture the features in the data. Based on the results of Fig.\ref{Fig.8/obj}, we found that the ACC value performs best when the $\epsilon$ value is between 0.5 and 0.8. Therefore, in subsequent experiments, we narrow the range of $\epsilon$ to between 0.5 and 0.8 to more accurately search for the optimal parameters.

Through the visual analysis and adjustment of the parameter value ranges mentioned above, we have determined more reasonable parameter search intervals for the models involved in the experiment. This will help improve the model's performance and generalization ability in subsequent experiments, allowing for better handling of different types of datasets.
\begin{figure}[H]
\begin{minipage}{0.3\textwidth}
  \centerline{\includegraphics[trim = 0mm 0mm 0mm 0mm, clip=true, width=5.5cm]{Australian11.eps}}
  \centerline{(a) Australian}
\end{minipage}
\hfill
\begin{minipage}{0.3\textwidth}
  \centerline{\includegraphics[trim = 0mm 0mm 0mm 0mm, clip=true, width=5.5cm]{Detecting11.eps}}
  \centerline{(b) Detecting}
\end{minipage}
\hfill
\begin{minipage}{0.3\textwidth}
  \centerline{\includegraphics[trim = 0mm 0mm 0mm 0mm, clip=true, width=5.5cm]{Hepat11.eps}}
  \centerline{(c) Hepatitis}
\end{minipage}
\begin{minipage}{0.3\textwidth}
  \centerline{\includegraphics[trim = 0mm 0mm 0mm 0mm, clip=true, width=5.5cm]{Ionosp11.eps}}
  \centerline{(d) Ionosphere}
\end{minipage}
\hfill
\begin{minipage}{0.3\textwidth}
  \centerline{\includegraphics[trim = 0mm 0mm 0mm 0mm, clip=true, width=5.5cm]{qsar11.eps}}
  \centerline{(e) QSAR}
\end{minipage}
\hfill
\begin{minipage}{0.3\textwidth}
  \centerline{\includegraphics[trim = 0mm 0mm 0mm 0mm, clip=true, width=5.5cm]{WDBC11.eps}}
  \centerline{(f) WDBC}
\end{minipage}
\caption{Variation Diagram of Parameter $C_{i}(i=1,2,3,4)$ and Accuracy in The Proposed GAHTSVM.}
\label{Fig.9/obj}
\end{figure}

\begin{figure}[H]
\begin{minipage}{0.3\textwidth}
  \centerline{\includegraphics[trim = 0mm 0mm 0mm 0mm, clip=true, width=5.5cm]{Australianzhuxing.eps}}
  \centerline{(a) Australian}
\end{minipage}
\hfill
\begin{minipage}{0.3\textwidth}
  \centerline{\includegraphics[trim = 0mm 0mm 0mm 0mm, clip=true, width=5.5cm]{Detectingzhuxing.eps}}
  \centerline{(b) Detecting}
\end{minipage}
\hfill
\begin{minipage}{0.3\textwidth}
  \centerline{\includegraphics[trim = 0mm 0mm 0mm 0mm, clip=true, width=5.5cm]{hepatzhuxing.eps}}
  \centerline{(c) Hepatitis}
\end{minipage}
\begin{minipage}{0.3\textwidth}
  \centerline{\includegraphics[trim = 0mm 0mm 0mm 0mm, clip=true, width=5.5cm]{Ionospherezhuxing.eps}}
  \centerline{(d) Ionosphere}
\end{minipage}
\hfill
\begin{minipage}{0.3\textwidth}
  \centerline{\includegraphics[trim = 0mm 0mm 0mm 0mm, clip=true, width=5.5cm]{QSARzhuxing.eps}}
  \centerline{(e) QSAR}
\end{minipage}
\hfill
\begin{minipage}{0.3\textwidth}
  \centerline{\includegraphics[trim = 0mm 0mm 0mm 0mm, clip=true, width=5.5cm]{WDBCzhuxing.eps}}
  \centerline{(f) WDBC}
\end{minipage}
\caption{Variation Diagram of Parameter $\epsilon$ and Accuracy in The Proposed GAHTSVM.}
\label{Fig.8/obj}
\end{figure}

\begin{table}[H]
\caption{Performance Comparison of Eight Nonlinear Algorithms on 15 Datasets (\(r = 0 \)).}
\label{N1}
\scriptsize
\begin{center}
\begin{tabular}{lcccccccc}
\toprule
& \textbf{TSVM} & \textbf{LSTSVM} & \textbf{HTSVM} & \textbf{CPin-GTBSVM} & \textbf{CNN-SVM} & \textbf{Wave-TSVM} & \textbf{SSVM} & \textbf{GAHTSVM} \\
\noalign{\smallskip}\hline\noalign{\smallskip}
Datasets & ACC(\%) & ACC(\%) & ACC(\%) & ACC(\%) & ACC(\%) & ACC(\%) & ACC(\%) & ACC(\%) \\
 & Times (s) & Times (s) & Times (s) & Times (s) & Times (s) & Times (s) & Times (s) & Times (s) \\
 & Rank & Rank & Rank & Rank & Rank & Rank & Rank & Rank \\
\noalign{\smallskip}\midrule
a8a & 74.70 & 77.59 & 78.95 & 74.96 & 76.62 & 72.43 & 75.93 & 78.52 \\
 & 183.29 & 38.2507 & 173.6755 & 187.3442 & 183.0975 & 165.4306 & 2.263 & 178.7633 \\
 & 7 & 3 & 1 & 6 & 4 & 8 & 5 & 2 \\
Australian & 86.59 & 86.94 & 86.81 & 87.68 & 87.68 & 87 & 87.14 & 87.82 \\
 & 0.28 & 0.1352 & 1.0058 & 0.1399 & 0.4211 & 0.8249 & 0.7342 & 0.7958 \\
 & 8 & 6 & 7 & 2 & 2 & 5 & 4 & 1 \\
Bank & 87.15 & 86.51 & 86.96 & 87.54 & 88.23 & 88.23 & 87.25 & 88.55 \\
 & 0.2022 & 0.1886 & 0.2187 & 0.2523 & 0.2134 & 0.2949 & 0.1096 & 0.1087 \\
 & 6 & 8 & 7 & 4 & 2 & 2 & 5 & 1 \\
Cancer & 94.21 & 94.48 & 94.71 & 94.58 & 95.26 & 94.52 & 94.06 & 95.48 \\
 & 0.5283 & 0.0771 & 0.4049 & 0.6689 & 0.5257 & 0.4047 & 0.7244 & 0.7288 \\
 & 7 & 6 & 3 & 4 & 2 & 5 & 8 & 1 \\
Detecting & 99.94 & 96.34 & 97.08 & 98.53 & 97.13 & 96.59 & 97.65 & 100.00 \\
 & 3.2584 & 1.5724 & 4.0798 & 3.1715 & 4.9197 & 3.1269 & 1.5979 & 3.6082 \\
 & 2 & 8 & 6 & 3 & 5 & 7 & 4 & 1 \\
German & 75.30 & 74.30 & 75.99 & 76.70 & 77.80 & 76.55 & 75.28 & 77.60 \\
 & 1.5709 & 0.1842 & 1.5843 & 1.1775 & 1.6866 & 1.339 & 0.5906 & 1.6069 \\
 & 6 & 8 & 5 & 3 & 1 & 4 & 7 & 2 \\
Hepatitis & 78.67 & 76.67 & 76.43 & 78.98 & 80.24 & 79.47 & 80 & 82 \\
 & 0.789 & 0.0724 & 0.7742 & 0.5257 & 0.8133 & 0.69 & 0.1317 & 0.6745 \\
 & 6 & 7 & 8 & 5 & 2 & 4 & 3 & 1 \\
Ionosphere & 87.18 & 87.12 & 86.54 & 88.24 & 86.13 & 87.5 & 87.91 & 87.82 \\
 & 0.2989 & 0.059 & 0.2435 & 0.3537 & 0.363 & 0.2573 & 0.1943 & 0.4002 \\
 & 5 & 6 & 7 & 1 & 8 & 4 & 2 & 3 \\
mushroom & 66.42 & 66.73 & 66.17 & 66.25 & 68.26 & 63.36 & 67.44 & 68.77 \\
 & 190.7755 & 50.1695 & 227.7713 & 236.8009 & 221.8231 & 176.4429 & 2.5481 & 240.8424 \\
 & 5 & 4 & 7 & 6 & 2 & 8 & 3 & 1 \\
QSAR & 78.37 & 83.56 & 80.64 & 81.37 & 83 & 82 & 82.1 & 84.71 \\
 & 0.7835 & 0.1604 & 1.0625 & 1.223 & 1.5411 & 0.7462 & 0.6237 & 1.4657 \\
 & 8 & 2 & 7 & 6 & 3 & 5 & 4 & 1 \\
Spam & 92.31 & 89.76 & 90.02 & 92.04 & 92.94 & 88.98 & 91.58 & 92.06 \\
 & 0.6861 & 0.2541 & 0.7073 & 0.2527 & 0.3638 & 0.5231 & 0.0794 & 0.9592 \\
 & 2 & 7 & 6 & 4 & 1 & 8 & 5 & 3 \\
Spectf & 78.23 & 78 & 76.25 & 77.67 & 78.42 & 82.42 & 78.77 & 81.54 \\
 & 0.446 & 0.0565 & 0.2078 & 0.3376 & 0.2341 & 0.3907 & 0.1125 & 0.1605 \\
 & 5 & 6 & 8 & 7 & 4 & 1 & 3 & 2 \\
Vote & 95.71 & 95.46 & 95.32 & 95.42 & 94.96 & 96.15 & 96 & 95.10 \\
 & 1.1432 & 0.0585 & 0.7542 & 0.0749 & 0.2387 & 0.9898 & 0.2347 & 0.3134 \\
 & 3 & 4 & 6 & 5 & 8 & 1 & 2 & 7 \\
WDBC & 96.32 & 96.52 & 96.37 & 95.78 & 95.95 & 96.78 & 96 & 96.75 \\
 & 0.8753 & 0.0851 & 1.8166 & 1.0538 & 0.0519 & 0.72 & 0.3936 & 0.6389 \\
 & 5 & 3 & 4 & 8 & 7 & 1 & 6 & 2 \\
Wholesale & 83.26 & 84.19 & 83.72 & 83.77 & 85.43 & 85.26 & 83.61 & 86.19 \\
 & 1.244 & 0.0525 & 0.9172 & 1.2213 & 0.9842 & 1.1901 & 0.5451 & 0.9255 \\
 & 8 & 4 & 6 & 5 & 2 & 3 & 7 & 1 \\
\noalign{\smallskip}\bottomrule
\end{tabular}
\end{center}
\end{table}

%%%%%%%%%%%%%%%%%%%
\begin{table}[H]
\caption{Performance Comparison of Eight Nonlinear Algorithms on 15 Datasets (\(r = 0.1 \)).}
\label{N2}
\scriptsize
\begin{center}
\begin{tabular}{lcccccccc}
\toprule
& \textbf{TSVM} & \textbf{LSTSVM} & \textbf{HTSVM} & \textbf{CPin-GTBSVM} & \textbf{CNN-SVM} & \textbf{Wave-TSVM} & \textbf{SSVM} & \textbf{GAHTSVM} \\
\noalign{\smallskip}\hline\noalign{\smallskip}
Datasets & ACC(\%) & ACC(\%) & ACC(\%) & ACC(\%) & ACC(\%) & ACC(\%) & ACC(\%) & ACC(\%) \\
 & Times (s) & Times (s) & Times (s) & Times (s) & Times (s) & Times (s) & Times (s) & Times (s) \\
 & Rank & Rank & Rank & Rank & Rank & Rank & Rank & Rank \\
\noalign{\smallskip}\midrule
a8a & 74.05 & 76.4 & 76.2 & 73.93 & 75.82 & 71.98 & 75.55 & 75.74 \\
 & 211.8237 & 65.8551 & 176.1381 & 139.9598 & 229.9884 & 187.2236 & 2.267 & 178.8989 \\
 & 6 & 1 & 2 & 7 & 3 & 8 & 5 & 4 \\
Australian & 85.71 & 85.71 & 84.72 & 86.81 & 87.39 & 86.43 & 86.52 & 86.85 \\
 & 0.7942 & 0.0689 & 0.7693 & 0.7207 & 0.8933 & 0.8533 & 0.7596 & 0.8017 \\
 & 6 & 6 & 8 & 3 & 1 & 5 & 4 & 2 \\
Bank & 86.64 & 86.93 & 85.78 & 86.11 & 87.33 & 86.78 & 86.73 & 87.31 \\
 & 0.7961 & 0.2136 & 1.0625 & 0.762 & 0.8973 & 0.3759 & 0.1113 & 1.9803 \\
 & 6 & 3 & 8 & 7 & 1 & 4 & 5 & 2 \\
Cancer & 92.14 & 93.06 & 91.25 & 93.13 & 92.25 & 92.16 & 93.51 & 93.16 \\
 & 0.4356 & 0.0801 & 0.4836 & 0.4326 & 0.4579 & 0.4219 & 0.7506 & 0.7228 \\
 & 7 & 4 & 8 & 3 & 5 & 6 & 1 & 2 \\
Detecting & 98.93 & 95.88 & 95.84 & 96.81 & 97.72 & 96.23 & 97 & 99.97 \\
 & 3.0267 & 1.7672 & 5.9792 & 5.1143 & 4.1126 & 3.6533 & 1.6324 & 3.2213 \\
 & 2 & 7 & 8 & 5 & 3 & 6 & 4 & 1 \\
German & 74.6 & 73.7 & 75.42 & 74.6 & 76.31 & 75.42 & 74.39 & 75.6 \\
 & 0.5194 & 0.1514 & 1.0737 & 0.5199 & 0.5257 & 1.4207 & 0.6789 & 1.4685 \\
 & 5 & 8 & 3 & 5 & 1 & 3 & 7 & 2 \\
Hepatitis & 74.67 & 75.13 & 76.03 & 78.62 & 79.17 & 78.87 & 79.42 & 81.33 \\
 & 1.241 & 0.0519 & 1.2061 & 1.3931 & 1.4738 & 0.7643 & 0.1323 & 1.1202 \\
 & 8 & 7 & 6 & 5 & 3 & 4 & 2 & 1 \\
Ionosphere & 86.94 & 85.86 & 85.22 & 87.91 & 86 & 86.52 & 87.4 & 87.53 \\
 & 0.5725 & 0.0692 & 0.3657 & 0.4576 & 0.4955 & 0.3998 & 0.2021 & 0.1719 \\
 & 4 & 7 & 8 & 1 & 6 & 5 & 3 & 2 \\
mushroom & 65.07 & 65.56 & 65.27 & 65.74 & 67.66 & 61.27 & 65.88 & 68.16 \\
 & 222.0515 & 76.1603 & 231.4292 & 244.7983 & 247.801 & 195.0609 & 2.6782 & 245.3363 \\
 & 7 & 5 & 6 & 4 & 2 & 8 & 3 & 1 \\
QSAR & 78.08 & 82.54 & 79.55 & 80.9 & 81.17 & 81.36 & 81.18 & 81.25 \\
 & 0.5833 & 0.144 & 3.0462 & 0.9822 & 0.7862 & 0.7931 & 0.6799 & 1.3033 \\
 & 8 & 1 & 7 & 6 & 5 & 2 & 4 & 3 \\
Spam & 89.56 & 89.42 & 89.35 & 89.81 & 89.03 & 88.44 & 90.6 & 89.4 \\
 & 0.7187 & 0.0513 & 0.3176 & 0.6011 & 0.4917 & 0.5231 & 0.0777 & 0.3677 \\
 & 3 & 4 & 6 & 2 & 7 & 8 & 1 & 5 \\
Spectf & 76.23 & 77.15 & 75.11 & 76.09 & 77.9 & 80.63 & 77.32 & 78.38 \\
 & 0.2807 & 0.0528 & 0.1978 & 0.3874 & 0.4531 & 0.3811 & 0.13 & 0.1837 \\
 & 6 & 5 & 8 & 7 & 3 & 1 & 4 & 2 \\
Vote & 94.52 & 92.09 & 94.96 & 93.8 & 94.12 & 95.24 & 95.18 & 93.33 \\
 & 1.3135 & 0.0616 & 0.3425 & 0.3943 & 0.2576 & 1.18 & 0.293 & 0.2325 \\
 & 4 & 8 & 3 & 6 & 5 & 1 & 2 & 7 \\
WDBC & 95.79 & 95.02 & 95.64 & 95.38 & 95.21 & 95.11 & 95.24 & 96.07 \\
 & 1.2888 & 0.061 & 1.2541 & 1.3576 & 1.2918 & 0.8864 & 0.4611 & 1.3937 \\
 & 2 & 8 & 3 & 4 & 6 & 7 & 5 & 1 \\
Wholesale & 82.56 & 82.5 & 82.63 & 83.77 & 82.91 & 84.73 & 82.01 & 83.13 \\
 & 1.869 & 0.0594 & 1.1367 & 1.5357 & 1.4588 & 1.2502 & 0.636 & 1.26 \\
 & 6 & 7 & 5 & 2 & 4 & 1 & 8 & 3 \\
\noalign{\smallskip}\bottomrule
\end{tabular}
\end{center}
\end{table}

%%%%%%%%%
\begin{table}[H]
\caption{Performance Comparison of Eight Nonlinear Algorithms on 15 Datasets (\(r = 0.3 \)).}
\label{N3}
\scriptsize
\begin{center}
% Resize the table to fit within the page width if necessary
%\resizebox{\textwidth}{!}{
\begin{tabular}{lcccccccc}
\toprule
 & \textbf{TSVM} & \textbf{LSTSVM} & \textbf{HTSVM} & \textbf{CPin-GTBSVM} & \textbf{CNN-SVM} & \textbf{Wave-TSVM} & \textbf{SSVM} & \textbf{GAHTSVM} \\
\noalign{\smallskip}\hline\noalign{\smallskip}
Datasets & ACC(\%) & ACC(\%) & ACC(\%) & ACC(\%) & ACC(\%) & ACC(\%) & ACC(\%) & ACC(\%) \\
         & Times (s) & Times (s) & Times (s) & Times (s) & Times (s) & Times (s) & Times (s) & Times (s) \\
         & Rank & Rank & Rank & Rank & Rank & Rank & Rank & Rank \\
\noalign{\smallskip}\midrule
a8a      & 70.05 & 75.79 & 73.23 & 71.75 & 74.36 & 70.65 & 74.88 & 75.52 \\
         & 212.81 & 74.09 & 346.80 & 552.84 & 357.80 & 198.22 & 2.33 & 438.81 \\
         & 8     & 1     & 5     & 6     & 4     & 7     & 3     & 2     \\
Australian & 84.59 & 85.07 & 84.22 & 85.22 & 84.78 & 85.24 & 85.98 & 85.74 \\
         & 0.68  & 0.07  & 0.79  & 0.65  & 0.89  & 0.88  & 0.79  & 1.05  \\
         & 7     & 5     & 8     & 4     & 6     & 3     & 1     & 2     \\
Bank     & 81.79 & 80.07 & 81.19 & 82.28 & 81.77 & 82.37 & 83.10 & 82.89 \\
         & 0.52  & 0.19  & 1.11  & 0.58  & 0.92  & 0.45  & 0.12  & 1.71  \\
         & 5     & 8     & 7     & 4     & 6     & 3     & 1     & 2     \\
Cancer   & 89.58 & 89.12 & 89.66 & 88.76 & 89.39 & 91.10 & 92.37 & 92.12 \\
         & 0.70  & 0.08  & 0.43  & 0.99  & 0.82  & 0.54  & 0.65  & 0.64  \\
         & 5     & 7     & 4     & 8     & 6     & 3     & 1     & 2     \\
Detecting & 95.23 & 95.65 & 92.70 & 94.84 & 96.20 & 95.00 & 96.23 & 96.97 \\
         & 3.72  & 1.89  & 6.55  & 5.71  & 4.68  & 3.71  & 1.65  & 3.66  \\
         & 5     & 4     & 8     & 7     & 3     & 6     & 2     & 1     \\
German   & 71.10 & 68.54 & 72.28 & 72.31 & 72.68 & 73.83 & 73.12 & 71.80 \\
         & 2.19  & 0.13  & 1.23  & 1.06  & 1.78  & 1.55  & 0.78  & 1.62  \\
         & 7     & 8     & 5     & 4     & 3     & 1     & 2     & 6     \\
Hepatitis & 72.67 & 70.56 & 74.93 & 73.37 & 74.22 & 74.56 & 75.36 & 74.67 \\
         & 1.12  & 0.05  & 1.41  & 1.37  & 1.27  & 0.90  & 0.14  & 1.13  \\
         & 7     & 8     & 2     & 6     & 5     & 4     & 1     & 3     \\
Ionosphere & 82.71 & 82.70 & 82.19 & 84.71 & 82.98 & 84.21 & 84.01 & 83.65 \\
         & 0.43  & 0.06  & 0.47  & 0.48  & 0.12  & 0.42  & 0.21  & 0.47  \\
         & 6     & 7     & 8     & 1     & 5     & 2     & 3     & 4     \\
mushroom & 62.69 & 63.27 & 64.18 & 63.68 & 65.84 & 57.00 & 64.92 & 65.18 \\
         & 201.34 & 43.11 & 198.52 & 234.67 & 212.44 & 198.54 & 2.69  & 226.56 \\
         & 7     & 6     & 4     & 5     & 1     & 8     & 3     & 2     \\
QSAR     & 77.31 & 76.73 & 78.61 & 78.52 & 77.63 & 78.42 & 79.33 & 79.06 \\
         & 1.38  & 0.18  & 1.08  & 2.06  & 1.01  & 0.88  & 0.69  & 1.89  \\
         & 7     & 8     & 3     & 4     & 6     & 5     & 1     & 2     \\
Spam     & 87.50 & 87.13 & 86.19 & 87.22 & 88.04 & 87.26 & 88.00 & 88.00 \\
         & 0.92  & 0.05  & 0.34  & 0.63  & 0.55  & 0.70  & 0.10  & 0.39  \\
         & 4     & 7     & 8     & 6     & 1     & 5     & 2     & 2     \\
Spectf   & 74.83 & 75.77 & 73.10 & 73.23 & 74.55 & 75.00 & 74.62 & 74.23 \\
         & 0.31  & 0.06  & 0.12  & 0.34  & 0.26  & 0.35  & 0.15  & 0.27  \\
         & 3     & 1     & 8     & 7     & 5     & 2     & 4     & 6     \\
Vote     & 90.00 & 91.76 & 91.99 & 92.71 & 93.00 & 92.17 & 91.00 & 92.14 \\
         & 0.43  & 0.06  & 0.44  & 0.57  & 0.42  & 1.21  & 0.31  & 0.46  \\
         &8&6&5&2&1&3&7&4\\
WDBC     & 92.93 & 92.07 & 93.24 & 92.37 & 93.72 & 93.13 & 93.06 & 93.21 \\
         & 1.30  & 0.18  & 1.18  & 1.53  & 1.55  & 1.79  & 0.52  & 1.40  \\
         & 6     & 8     & 2     & 7     & 1     & 4     & 5     & 3     \\
Wholesale & 79.35 & 76.72 & 77.17 & 79.25 & 78.50 & 81.00 & 79.23 & 79.32 \\
         & 1.54  & 0.06  & 1.17  & 1.47  & 1.37  & 1.31  & 0.64  & 1.52  \\
         & 2     & 8     & 7     & 4     & 6     & 1     & 5     & 3     \\
\noalign{\smallskip}\bottomrule
\end{tabular}
%}
\end{center}
\end{table}
To evaluate the classification performance of the model, we will perform experimental tests on the eight algorithms on the real data set described above. The experimental results without noise and those with $10\%$ and $30\%$ Gaussian noise are shown in \autoref{N1}, \autoref{N2} and \autoref{N3}.

\autoref{N1}, \autoref{N2}, and \autoref{N3} show the results of eight nonlinear algorithms for \(r = 0 \), \(r = 0.1 \), and \(r = 0.3 \), respectively, on 15 datasets. All experimental results are based on the optimal parameters, and the classification accuracy is expressed as the average of ACC. The following is the result of the analysis of the performance of all algorithms.

In the scenarios without noise, GAHTSVM shows significant advantages. Out of 15 datasets, GAHTSVM ranked first in 8 datasets, including Australian, Bank, and Cancer. For example, with Detecting data set, GAHTSVM has an accuracy of 100\%, while for detecting data set, it is as high as 95.48\%. Compared with other algorithms, GAHTSVM's advantages in accuracy (ACC) and ranking reflect its stability in low noise environments and strong generalization ability. As the noise level increases to 10\%, the classification performance of GAHTSVM still shows strong advantages, although it fluctuates, ranking first in four datasets (such as Detecting and Hepatitis), and maintaining the second place in five datasets. For example, the ACC of Detecting dataset is still as high as 99.97\%, showing that it can still maintain excellent classification ability under high noise. Although ACC decreases on datasets such as Australian and Bank, GAHTSVM performs particularly well on the hepatitis dataset. By dynamically adjusting feature weights, its ACC reaches 81.33\%, which is ahead of other algorithms, demonstrating its adaptability and robustness in complex data environments. These results show that GAHTSVM can still maintain high classification performance under noise interference, especially in specific application scenarios, showing unique advantages. Facing 30\% strong noise, GAHTSVM ranks first only on the Detecting dataset (ACC 96.97\%), while its performance generally declines on other datasets. For example, on the German dataset, its rank drops from 2nd to 6th, and the ACC decreases from 77.6\% to 71.80\% as the noise is 0. Nevertheless, compared to the other 7 algorithms, GAHTSVM ranks second on 5 datasets, third on 3 datasets, and performs well on more than half of the datasets, indicating that GAHTSVM has strong noise resistance and can maintain a high classification accuracy even under noisy conditions.

According to \autoref{N1}, \autoref{N2} and \autoref{N3}, we calculate the average ranking and accuracy on 8 algorithms and 15 datasets, the results are shown in \autoref{N4}.
\begin{table}[H]
    \caption{Average Accuracy and Ranking of 8 Algorithm Models on Real-World Datasets with Varying Noise Levels}
    \label{N4}
    \scriptsize
    \centering
    \begin{tabular}{lcccccccc}
        \toprule
        & \textbf{TSVM} & \textbf{LSTSVM} & \textbf{HTSVM} & \textbf{CPin - GTBSVM} & \textbf{CNN - SVM} & \textbf{Wave - TSVM} & \textbf{SSVM} & \textbf{GAHTSVM} \\
        \midrule
        Avg.ACC 0\% & 84.96 & 84.94 & 84.8 & 85.3 & 85.87 & 85.15 & 85.38 & 86.86 \\
        Avg.rank 0\%& 5.53 & 5.47 & 5.87 & 4.6 & 3.53 & 4.40 & 4.53 & 1.93 \\
        \hline
        Avg.ACC 10\%& 83.7 & 83.8 & 83.53 & 84.23 & 84.67 & 84.08 & 84.53 & 85.15 \\
        Avg.rank 10\%& 5.33 & 5.4 & 5.93 & 4.47 & 3.67 & 4.6 & 3.87 & 2.53 \\
        \hline
        Avg.ACC 30\%& 80.82 & 80.73 & 80.99 & 81.35 & 81.84 & 81.4 & 82.35 & 82.30 \\
        Avg.rank 30\%& 5.8 & 6.13 & 5.6 & 5 & 3.93 & 3.8 & 2.73 & 2.93\\
        \bottomrule
    \end{tabular}
\end{table}

Overall, GAHTSVM performs well at different noise levels. When the noise level is 0, the average accuracy reaches 86.86, which is the highest value among the 8 models, and the average ranking is 1.93, which is the best performance. When the noise level was increased to 10\%, Avg. ACC remained at 85.15 and Avg. rank at 2.53, which continued to rank the top. When the noise level was further increased to 30\%, Avg.ACC dropped to 82.30 and Avg.rank to 2.93, but still maintained good performance. It can be seen that GAHTSVM can maintain high accuracy and superior ranking in all kinds of noise scenarios, especially in the noise-free environment, showing obvious advantages, overall highlighting its excellent comprehensive performance and noise robustness.
\begin{figure}[H]
    \begin{minipage}{0.45\textwidth}
        \centerline{\includegraphics[trim = 0mm 0mm 0mm 0mm, clip=true, width=9.5cm]{0noise_zhuxing.eps}}
        \centerline{(a) Accuracy of 8 Algorithms without Noise.}
    \end{minipage}
    \hfill
    \begin{minipage}{0.45\textwidth}
        \centerline{\includegraphics[trim = 0mm 0mm 0mm 0mm, clip=true, width=9.5cm]{10noise_zhuxing.eps}}
        \centerline{(b) Accuracy of 8 Algorithms with 10\% Noise.}
    \end{minipage}
    \vspace{0.3cm}
    \begin{center}
        \begin{minipage}{0.45\textwidth}
            \centerline{\includegraphics[trim = 0mm 0mm 0mm 0mm, clip=true, width=9.5cm]{30noise_zhuxing.eps}}
            \centerline{(c) Accuracy of 8 Algorithms with 30\% Noise.}
        \end{minipage}
    \end{center}
    \caption{ Accuracy of 8 Algorithms with Varying Noise Levels}
    \label{Fig.5/obj}
\end{figure}
Fig.\ref{Fig.5/obj} provides a more intuitive view of how the eight algorithms perform across 15 real datasets with varying levels of noise.
\subsection{Friedman Test}
We evaluated eight algorithms using both real and synthetic datasets, and the results show that GAHTSVM achieves relatively high accuracy. To thoroughly analyze the differences in classification performance among the eight algorithms across datasets, we conducted a Friedman test for statistical analysis \cite{ref34}. As a non-parametric test, the Friedman test is well-suited for comparing multiple related samples and is widely preferred in machine learning performance evaluation due to its simplicity and robustness. The core principle of the test is to assess performance differences by analyzing the ranking data of algorithms across different datasets.

The null hypothesis \(H_{0}\) for this test posits that there are no significant differences in the classification performance of the eight algorithms, while the alternative hypothesis \(H_{1}\) suggests that significant differences do exist. The procedure involves ranking the algorithms based on their accuracy across different datasets and then calculating the Friedman statistic \(F_F\), which integrates the ranking differences among the algorithms. By comparing \(F_F\) with the critical value corresponding to the degrees of freedom, if \(F_F\) exceeds the critical value, the null hypothesis is rejected, indicating significant performance differences between the algorithms.

Although GAHTSVM showed strong overall performance in the preliminary numerical experiments, its results were not always optimal across all datasets. Therefore, the Friedman test, as a rigorous statistical method, is crucial to validate the significance of GAHTSVM's performance advantage. Through this test, we can scientifically determine whether there are statistically significant differences in classification performance between GAHTSVM and the other seven algorithms, thus providing strong theoretical support for GAHTSVM's effectiveness.

The Friedman statistic is calculated as follows:
\begin{eqnarray}\label{62}
\chi^{2}=\frac{12N}{p(p+1)}\left[\sum_{j}R_{j}^{2}-\frac{p(p+1)^{2}}{4}\right],
\end{eqnarray}
where \(N\) represents the number of datasets used in the experiments, \(p\) is the number of algorithms employed, and \(R_{j}\) denotes the average ranking of the \(j\)-th algorithm across all datasets. The key term in the computation is \(\sum_{j}R_{j}^{2} - \frac{p(p+1)^{2}}{4}\), which consists of:
\(\sum_{j}R_{j}^{2}\), the sum of the squared average rankings of all algorithms, reflecting the degree of variability in the algorithm rankings, and
\(\frac{p(p+1)^{2}}{4}\), the theoretical expectation under the null hypothesis (i.e., assuming no significant difference between the algorithms).

The \(F\)-distribution with degrees of freedom \(p-1\) and \((p-1)(N-1)\) is used to calculate the following:
\begin{eqnarray}\label{63}
F_{F}=\frac{(N-1)\chi^{2}}{N(p-1)-\chi^{2}}.
\end{eqnarray}

The numerator's degrees of freedom, \((p-1)\), reflect the dimensions of the differences between the algorithms. The denominator��s degrees of freedom, \((p-1)(N-1)\), combine the number of algorithms and datasets to represent the error component of the overall data variation after accounting for algorithmic effects. This allows for the measurement of the impact of random factors on the results. After calculating \(F_F\), it is compared to the critical value for the corresponding degrees of freedom from the \(F\)-distribution table. If \(F_F\) exceeds the critical value, the null hypothesis of 'no significant difference between algorithms' is rejected, indicating that there are statistically significant differences in classification performance among the algorithms on the datasets.

In our experiment, $N=15$ and $p=8$, according to the equations \eqref{62} and \eqref{63}, when the noise is 0, 0.1 and 0.3, we get the values of $\chi^{2}$ are 25.5, 16.8 and 29 respectively, and the values of $F_{F}$ are 4.4983, 2.673 and 5.3449 respectively. In exploratory studies, a higher value of $\alpha$ (such as $\alpha = 0.1$) can increase the power of the test, making it easier to detect potential differences between algorithms. At a significance level of $\alpha = 0.1$, the critical value for $F(7,98)$ is $F_{\alpha} = 1.89$. Apparently,
\begin{eqnarray*}
F_{F}=4.4983>1.89=F_{\alpha},\\
F_{F}=2.673>1.89=F_{\alpha},\\
F_{F}=5.3449>1.89=F_{\alpha}.
\end{eqnarray*}

Therefore, reject null hypothesis $H_ {0}$, and accept the alternative hypothesis $H_ {1}$. In other words, there are significant differences among the seven comparison algorithms.

In order to further compare the performance of the eight algorithms, we used the Bonferroni-Dunn post hoc test to assess the significant differences between the algorithms. Based on the Studentized range statistic divided by $\sqrt{2}$, we set $q_{\alpha=0.1}(p) = 2.95$ and derive the critical difference statistic (CD):
\begin{eqnarray}\label{64}
CD = q_{\alpha=0.1}(p) \times \sqrt{\frac{p(p+1)}{6N}} = 2.95 \times \sqrt{\frac{8 \times 9}{6 \times 15}} = 2.6386
\end{eqnarray}

When the difference between the calculated rankings of the two algorithms exceeds a critical value, it indicates that there is a significant difference between the two algorithms. In order to further verify this result, we show the average ranking comparison table of GAHTSVM and other seven algorithms under different noise conditions and the post hoc tests figure.
\begin{table}[htbp]
\scriptsize
 \centering
    \caption{Difference of Average Ranks on 15 Datasets.}
    \label{CD}
    \begin{tabular}{lccc}
        \toprule
        Model comparison & \multicolumn{3}{c}{Gaussian noise} \\
        \cmidrule(lr){2-4}
        & r = 0 & r = 0.1 & r = 0.3 \\
        \midrule
        d(TSVM - GAHTSVM) & 3.6 & 2.8 & 2.87 \\
        d(LSTSVM - GAHTSVM) & 3.53 & 2.87 & 3.2 \\
        d(HTSVM - GAHTSVM) & 3.93 & 3.4 & 2.67 \\
        d(CPin - GTBSVM - GAHTSVM) & 2.67 & 1.93 & 2.07 \\
        d(CNN - SVM - GAHTSVM) & 1.6 & 1.13 & 1 \\
        d(Wave - SVM - GAHTSVM) & 2.47 & 2.07 & 0.86 \\
        d(SSVM - GAHTSVM)& 2.6 & 1.33 & -0.2 \\
        \bottomrule
    \end{tabular}
\end{table}

\begin{figure}[H]
    \begin{minipage}{0.45\textwidth}
        \centerline{\includegraphics[trim = 0mm 0mm 0mm 0mm, clip=true, width=9cm]{CD_0.eps}}
        \centerline{(a) The Post Hoc Test at r=0.}
    \end{minipage}
    \hfill
    \begin{minipage}{0.45\textwidth}
        \centerline{\includegraphics[trim = 0mm 0mm 0mm 0mm, clip=true, width=9cm]{CD_01.eps}}
        \centerline{(b) The Post Hoc Test at r=0.1.}
    \end{minipage}
    \vspace{0.3cm}
    \begin{center}
        \begin{minipage}{0.45\textwidth}
            \centerline{\includegraphics[trim = 0mm 0mm 0mm 0mm, clip=true, width=9cm]{CD_03.eps}}
            \centerline{(c)The Post Hoc Test at r=0.3.}
        \end{minipage}
    \end{center}
    \caption{Visualization of Post Hoc Tests for Real-Word Datesets.}
    \label{Fig.6/obj}
\end{figure}

Based on \autoref{CD} and Fig.\ref{Fig.6/obj} , The performance of GAHTSVM is significantly better than that of the other seven algorithms. However, when the noise level reaches 30\%, the dot position of GAHTSVM drops from the first to the second, and SSVM ranks the first at this time, but the difference between the two algorithms is very small. This fully proves that no matter what the noise level is, the performance difference between GAHTSVM and other algorithms is significantly more than the critical difference, with significant statistical significance, and its performance is always better than the other eight algorithms, which verifies the excellent performance and effectiveness of GAHTSVM in different noise environments.

\section{Conclusion}\label{s4}
In this paper, a novel generalized adaptive Huber loss function is proposed, and the GAHTSVM model based on this loss function is constructed. The outstanding advantage of the loss function is its excellent noise robustness, which can effectively suppress the adverse impact of noisy data points on the decision function by dynamically adjusting the robustness parameter $s$. At the same time, it innovatively introduces the insensitive region design, which significantly improves the sparsity of the model. By fusing the dominant features of the $\epsilon$-insensitive loss and the non-convex least squares loss, the proposed method further enhances the adaptability of the model to complex data distributions while maintaining the robustness of the traditional Huber loss. Theoretical analysis shows that the loss function achieves an ideal balance between noise control and model generalization, and provides a new solution for pattern recognition problems in high noise environment.

Experiments show that the GAHTSVM algorithm performs very well in complex data scenarios. In the experiment of adding Gaussian noise to real data sets, GAHTSVM ranked first in 8 out of 15 data sets without adding noise, showing its stability and generalization ability under low noise environment.When the noise increases to 10\%, the algorithm keeps ahead in some data sets by dynamically adjusting the feature weight, which reflects its adaptability and robustness in complex environments. In the face of 30\% strong noise, despite fluctuations, it is still outstanding on some data sets, and more than half of the rankings are in the forefront, showing strong anti-noise ability. In the scenario of adding outliers to artificial data sets, as shown in Fig.\ref{Fig.art/obj}, GAHTSVM is significantly superior to other algorithms and can effectively deal with the interference of outliers, highlighting its robustness and classification advantages.In conclusion, GAHTSVM can effectively overcome the interference of outliers on classification results, especially when the adaptive parameter \(s\) is large. By increasing the penalty for outliers, the adaptability of the model to abnormal data is further improved, and it shows excellent classification performance under the challenges of noise and outliers.

Although the proposed algorithm shows excellent performance in classification tasks, there is still room for optimization in the parameter selection process, especially in the solution efficiency. The future research will focus on the following directions: firstly, for the mechanism of adaptive parameter \(s \) in this paper, although parameter \(s \) can be automatically selected through this mechanism in model construction, the mechanism still relies on human Settings, and the parameters involved need to be manually adjusted, which may lead to waste in time and may fail to achieve the optimal effect. We hope that the optimal parameter selection can be fully automated on the basis of minimizing the calculation cost. Secondly, we will focus on exploring better models to effectively reduce the calculation time and further improve the classification effect of different data sets under the premise of ensuring the classification accuracy. Thirdly, the application of the model in this paper is extended to more diverse work scenarios such as multi-classification problems. In addition, subsequent research can deeply consider the adaptation of diverse models to practical hot application scenarios such as face recognition, fingerprint recognition, and UAV scheduling. It is worth emphasizing that the development of more innovative algorithms for GAHTSVM will also become a key focus of future research.

\section*{Author Contributions}
{\bf TianTian Jiang,} {\bf Guolin Yu \&} {\bf Jun Ma :} The responsibilities included algorithm development, software creation, numerical example preparation, original draft writing, and review and editing of the manuscript. All authors contributed equally to this work and have read and approved the final version
of the manuscript for publication.

\section*{Use of AI Tools Declaration}
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

\section*{Acknowledgments (All Sources of Funding of The Study Must Be Disclosed)}
This work was supported in part by the Fundamental Research Funds for the Central Universities of North Minzu University ( No.2023ZRLG01), in part by the National Natural Science Foundation of China (No.62366001, No.12361062), in part by the Key Research and Development Program of Ningxia (Introduction of Talents Project)(No. 2022BSB03046), in part by the Natural Science Foundation of Ningxia Provincial ( No.2023AAC02053) and in part by the Postgraduate Innova- tion Project of North Minzu University (YCX24261).
\section*{Conflict of Interest}
The authors declare that they have no conflict of interest and no relevant financial or non-financial interests to disclose.

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\end{document}
