In recent years, the random vector functional link (RVFL) network has gained significant popularity in hyperspectral image (HSI) classification due to its simplicity, speed, and strong generalization performance. However, despite these advantages, RVFL models face several limitations, particularly in handling non-linear relationships and complex data structures. The random initialization of input-to-hidden weights can lead to instability, and the model struggles with determining the optimal number of hidden nodes, affecting its performance on more challenging datasets. To address these issues, we propose a novel randomized based restricted kernel machine ($R^2KM$) model that combines the strehyperngths of RVFL and restricted kernel machines (RKM). $R^2KM$ introduces a layered structure that represents kernel methods using both visible and hidden variables, analogous to the energy function in restricted Boltzmann machines (RBM). This structure enables $R^2KM$ to capture complex data interactions and non-linear relationships more effectively, improving both interpretability and model robustness. A key contribution of $R^2KM$ is the introduction of a novel conjugate feature duality based on the Fenchel-Young inequality, which expresses the problem in terms of conjugate dual variables and provides an upper bound on the objective function. This duality enhances the model's flexibility and scalability, offering a more efficient and flexible solution for complex data analysis tasks. Extensive experiments on hyperspectral image datasets and real-world data from the UCI and KEEL repositories show that $R^2KM$ outperforms baseline models, demonstrating its effectiveness in classification and regression tasks.

Restricted kernel machines (RKMs) have considerably improved generalization in machine learning. Recent advancements explored various techniques within the RKM framework, integrating kernel functions with least squares support vector machines (LSSVM) to mirror the energy function of restricted Boltzmann machines (RBM), leading to enhanced performance. However, RKMs may face challenges in generalization when dealing with unevenly distributed or complexly clustered data. Additionally, as the dataset size increases, the computational burden of managing high-dimensional feature spaces can become substantial, potentially hindering performance in large-scale datasets. To address these challenges, we propose twin restricted kernel machine (TRKM). TRKM combines the benefits of twin models with the robustness of the RKM framework to enhance classification and regression tasks. By leveraging the Fenchel-Young inequality, we introduce a novel conjugate feature duality, allowing the formulation of classification and regression problems in terms of dual variables. This duality provides an upper bound to the objective function of the TRKM problem, resulting in a new methodology under the RKM framework. The model uses an energy function similar to that of RBM, incorporating both visible and hidden variables corresponding to both classes. Additionally, the kernel trick is employed to map data into a high-dimensional feature space, where the model identifies an optimal separating hyperplane using a regularized least squares approach. Experiments on UCI and KEEL datasets confirm TRKM's superiority over baselines, showcasing its robustness and efficiency in handling complex data. Furthermore, We implemented the TRKM model on the brain age dataset, demonstrating its efficacy in predicting brain age.

Restricted kernel machines (RKMs) have demonstrated a significant impact in enhancing generalization ability in the field of machine learning. Recent studies have introduced various methods within the RKM framework, combining kernel functions with the least squares support vector machine (LSSVM) in a manner similar to the energy function of restricted boltzmann machines (RBM), such that a better performance can be achieved. However, RKM's efficacy can be compromised by the presence of outliers and other forms of contamination within the dataset. These anomalies can skew the learning process, leading to less accurate and reliable outcomes. To address this critical issue and to ensure the robustness of the model, we propose the novel one-class RKM (OCRKM). In the framework of OCRKM, we employ an energy function akin to that of the RBM, which integrates both visible and hidden variables in a nonprobabilistic setting. The formulation of the proposed OCRKM facilitates the seamless integration of one-class classification method with the RKM, enhancing its capability to detect outliers and anomalies effectively. The proposed OCRKM model is evaluated over UCI benchmark datasets. Experimental findings and statistical analyses consistently emphasize the superior generalization capabilities of the proposed OCRKM model over baseline models across all scenarios.

In real-world applications, class-imbalanced datasets pose significant challenges for machine learning algorithms, such as support vector machines (SVMs), particularly in effectively managing imbalance, noise, and outliers. Fuzzy support vector machines (FSVMs) address class imbalance by assigning varying fuzzy memberships to samples; however, their sensitivity to imbalanced datasets can lead to inaccurate assessments. The recently developed slack-factor-based FSVM (SFFSVM) improves traditional FSVMs by using slack factors to adjust fuzzy memberships based on misclassification likelihood, thereby rectifying misclassifications induced by the hyperplane obtained via different error cost (DEC). Building on SFFSVM, we propose an improved slack-factor-based FSVM (ISFFSVM) that introduces a novel location parameter. This novel parameter significantly advances the model by constraining the DEC hyperplane's extension, thereby mitigating the risk of misclassifying minority class samples. It ensures that majority class samples with slack factor scores approaching the location threshold are assigned lower fuzzy memberships, which enhances the model's discrimination capability. Extensive experimentation on a diverse array of real-world KEEL datasets demonstrates that the proposed ISFFSVM consistently achieves higher F1-scores, Matthews correlation coefficients (MCC), and area under the precision-recall curve (AUC-PR) compared to baseline classifiers. Consequently, the introduction of the location parameter, coupled with the slack-factor-based fuzzy membership, enables ISFFSVM to outperform traditional approaches, particularly in scenarios characterized by severe class disparity. The code for the proposed model is available at \url{https://github.com/mtanveer1/ISFFSVM}.

One of the major difficulties in machine learning methods is categorizing datasets that are imbalanced. This problem may lead to biased models, where the training process is dominated by the majority class, resulting in inadequate representation of the minority class. Universum twin support vector machine (UTSVM) produces a biased model towards the majority class, as a result, its performance on the minority class is often poor as it might be mistakenly classified as noise. Moreover, UTSVM is not proficient in handling datasets that contain outliers and noises. Inspired by the concept of incorporating prior information about the data and employing an intuitionistic fuzzy membership scheme, we propose intuitionistic fuzzy universum twin support vector machines for imbalanced data (IFUTSVM-ID). We use an intuitionistic fuzzy membership scheme to mitigate the impact of noise and outliers. Moreover, to tackle the problem of imbalanced class distribution, data oversampling and undersampling methods are utilized. Prior knowledge about the data is provided by universum data. This leads to better generalization performance. UTSVM is susceptible to overfitting risks due to the omission of the structural risk minimization (SRM) principle in their primal formulations. However, the proposed IFUTSVM-ID model incorporates the SRM principle through the incorporation of regularization terms, effectively addressing the issue of overfitting. We conduct a comprehensive evaluation of the proposed IFUTSVM-ID model on benchmark datasets from KEEL and compare it with existing baseline models. Furthermore, to assess the effectiveness of the proposed IFUTSVM-ID model in diagnosing Alzheimer's disease (AD), we applied them to the Alzheimer's Disease Neuroimaging Initiative (ADNI) dataset. Experimental results showcase the superiority of the proposed IFUTSVM-ID models compared to the baseline models.

In the domain of machine learning, least square twin support vector machine (LSTSVM) stands out as one of the state-of-the-art models. However, LSTSVM suffers from sensitivity to noise and outliers, overlooking the SRM principle and instability in resampling. Moreover, its computational complexity and reliance on matrix inversions hinder the efficient processing of large datasets. As a remedy to the aforementioned challenges, we propose the robust granular ball LSTSVM (GBLSTSVM). GBLSTSVM is trained using granular balls instead of original data points. The core of a granular ball is found at its center, where it encapsulates all the pertinent information of the data points within the ball of specified radius. To improve scalability and efficiency, we further introduce the large-scale GBLSTSVM (LS-GBLSTSVM), which incorporates the SRM principle through regularization terms. Experiments are performed on UCI, KEEL, and NDC benchmark datasets; both the proposed GBLSTSVM and LS-GBLSTSVM models consistently outperform the baseline models.

Alzheimer's disease (AD) is a leading neurodegenerative condition and the primary cause of dementia, characterized by progressive cognitive decline and memory loss. Its progression, marked by shrinkage in the cerebral cortex, is irreversible. Numerous machine learning algorithms have been proposed for the early diagnosis of AD. However, they often struggle with the issues of noise, outliers, and class imbalance. To tackle the aforementioned limitations, in this article, we introduce a novel, robust, and flexible membership scheme called Flexi-Fuzz. This scheme integrates a novel flexible weighting mechanism, class probability, and imbalance ratio. The proposed flexible weighting mechanism assigns the maximum weight to samples within a specific proximity to the center, with a gradual decrease in weight beyond a certain threshold. This approach ensures that samples near the class boundary still receive significant weight, maintaining their influence in the classification process. Class probability is used to mitigate the impact of noisy samples, while the imbalance ratio addresses class imbalance. Leveraging this, we incorporate the proposed Flexi-Fuzz membership scheme into the least squares support vector machines (LSSVM) framework, resulting in a robust and flexible model termed Flexi-Fuzz-LSSVM. We determine the class-center using two methods: the conventional mean approach and an innovative median approach, leading to two model variants, Flexi-Fuzz-LSSVM-I and Flexi-Fuzz-LSSVM-II. To validate the effectiveness of the proposed Flexi-Fuzz-LSSVM models, we evaluated them on benchmark UCI and KEEL datasets, both with and without label noise. Additionally, we tested the models on the Alzheimer's Disease Neuroimaging Initiative (ADNI) dataset for AD diagnosis. Experimental results demonstrate the superiority of the Flexi-Fuzz-LSSVM models over baseline models.

Random vector functional link (RVFL), a variant of single-layer feedforward neural network (SLFN), has garnered significant attention due to its lower computational cost and robustness to overfitting. Despite its advantages, the RVFL network's reliance on the square error loss function makes it highly sensitive to outliers and noise, leading to degraded model performance in real-world applications. To remedy it, we propose the incorporation of the HawkEye loss (H-loss) function into the RVFL framework. The H-loss function features nice mathematical properties, including smoothness and boundedness, while simultaneously incorporating an insensitive zone. Each characteristic brings its own advantages: 1) Boundedness limits the impact of extreme errors, enhancing robustness against outliers; 2) Smoothness facilitates the use of gradient-based optimization algorithms, ensuring stable and efficient convergence; and 3) The insensitive zone mitigates the effect of minor discrepancies and noise. Leveraging the H-loss function, we embed it into the RVFL framework and develop a novel robust RVFL model termed H-RVFL. Notably, this work addresses a significant gap, as no bounded loss function has been incorporated into RVFL to date. The non-convex optimization of the proposed H-RVFL is effectively addressed by the Nesterov accelerated gradient (NAG) algorithm, whose computational complexity is also discussed. The proposed H-RVFL model's effectiveness is validated through extensive experiments on $40$ benchmark datasets from UCI and KEEL repositories, with and without label noise. The results highlight significant improvements in robustness and efficiency, establishing the H-RVFL model as a powerful tool for applications in noisy and outlier-prone environments.

In this paper, we propose enhanced feature based granular ball twin support vector machine (EF-GBTSVM). EF-GBTSVM employs the coarse granularity of granular balls (GBs) as input rather than individual data samples. The GBs are mapped to the feature space of the hidden layer using random projection followed by the utilization of a non-linear activation function. The concatenation of original and hidden features derived from the centers of GBs gives rise to an enhanced feature space, commonly referred to as the random vector functional link (RVFL) space. This space encapsulates nuanced feature information to GBs. Further, we employ twin support vector machine (TSVM) in the RVFL space for classification. TSVM generates the two non-parallel hyperplanes in the enhanced feature space, which improves the generalization performance of the proposed EF-GBTSVM model. Moreover, the coarser granularity of the GBs enables the proposed EF-GBTSVM model to exhibit robustness to resampling, showcasing reduced susceptibility to the impact of noise and outliers. We undertake a thorough evaluation of the proposed EF-GBTSVM model on benchmark UCI and KEEL datasets. This evaluation encompasses scenarios with and without the inclusion of label noise. Moreover, experiments using NDC datasets further emphasize the proposed model's ability to handle large datasets. Experimental results, supported by thorough statistical analyses, demonstrate that the proposed EF-GBTSVM model significantly outperforms the baseline models in terms of generalization capabilities, scalability, and robustness. The source code for the proposed EF-GBTSVM model, along with additional results and further details, can be accessed at https://github.com/mtanveer1/EF-GBTSVM.

On Efficient and Scalable Computation of the Nonparametric Maximum Likelihood Estimator in Mixture ModelsTwin support vector machine (TSVM) is an emerging machine learning model with versatile applicability in classification and regression endeavors. Nevertheless, TSVM confronts noteworthy challenges: $(i)$ the imperative demand for matrix inversions presents formidable obstacles to its efficiency and applicability on large-scale datasets; $(ii)$ the omission of the structural risk minimization (SRM) principle in its primal formulation heightens the vulnerability to overfitting risks; and $(iii)$ the TSVM exhibits a high susceptibility to noise and outliers, and also demonstrates instability when subjected to resampling. In view of the aforementioned challenges, we propose the granular ball twin support vector machine (GBTSVM). GBTSVM takes granular balls, rather than individual data points, as inputs to construct a classifier. These granular balls, characterized by their coarser granularity, exhibit robustness to resampling and reduced susceptibility to the impact of noise and outliers. We further propose a novel large-scale granular ball twin support vector machine (LS-GBTSVM). LS-GBTSVM's optimization formulation ensures two critical facets: $(i)$ it eliminates the need for matrix inversions, streamlining the LS-GBTSVM's computational efficiency, and $(ii)$ it incorporates the SRM principle through the incorporation of regularization terms, effectively addressing the issue of overfitting. The proposed LS-GBTSVM exemplifies efficiency, scalability for large datasets, and robustness against noise and outliers. We conduct a comprehensive evaluation of the GBTSVM and LS-GBTSVM models on benchmark datasets from UCI, KEEL, and NDC datasets. Our experimental findings and statistical analyses affirm the superior generalization prowess of the proposed GBTSVM and LS-GBTSVM models.