Multidimensional Datasets Research Articles

Carbon dioxide stripping from acidic wastewater through a vortex venturi tubular microbubble generator

Microbubble generation technology can improve CO2 stripping efficiency while consuming less sweeping gas. In this paper, a vortex venturi tubular microbubble generator is developed based on the concept of “Initial bubble generation with a thin plate type static mixing element” and “Bubble collapse with a venturi channel”. Microbubble characteristics were tested, followed by post-processing of the focused beam reflectance measurement (FBRM) to obtain a multidimensional dataset. The test results indicated that the microbubble quality generated by the vortex venturi tubular microbubble generator was higher than that of the SV-type static mixer. When the gas-liquid volume ratio was 1, the average bubble diameter was 107.05 μm, and microbubbles accounted for 93.29 %. The average bubble diameter of the SV-type static mixer was 473.55 μm, and the proportion of microbubbles was 23.81 % under identical conditions. Furthermore, the CO2 removal efficiency and pH enhancement rate were greatly affected by the gas-liquid volume ratio and the inlet water flow rate, and both fitted an exponential equation correlation. Accordingly, when the gas-liquid volume ratio increased, the quantity of bubbles generated significantly influenced the CO2 stripping efficiency. The stripping efficiency based on the tubular microbubble generator is significantly higher than that of the SV-type static mixer. When the gas-liquid volume ratio was 1, the CO2 removal efficiency was equivalent to that of the SV-type static mixer at a gas-liquid volume ratio of 3.5. When achieving equal CO2 removal efficiency, the gaseous nitrogen consumption of the tubular microbubble generator was 48.61 %-63.30 % lower than that of the SV-type static mixer, and the corresponding energy consumption of the gas compressor was reduced by 38.24 %-55.83 %. The CO2 stripping process based on the vortex venturi tubular microbubble generator offers an environmentally and economically viable treatment option for acid wastewater.

Rapid Identification of Drug Mechanisms with Deep Learning-Based Multichannel Surface-Enhanced Raman Spectroscopy.

Rapid identification of drug mechanisms is vital to the development and effective use of chemotherapeutics. Herein, we develop a multichannel surface-enhanced Raman scattering (SERS) sensor array and apply deep learning approaches to realize the rapid identification of the mechanisms of various chemotherapeutic drugs. By implementing a series of self-assembled monolayers (SAMs) with varied molecular characteristics to promote heterogeneous physicochemical interactions at the interfaces, the sensor can generate diversified SERS signatures for directly high-dimensionality fingerprinting drug-induced molecular changes in cells. We further train the convolutional neural network model on the multidimensional SAM-modulated SERS data set and achieve a discriminatory accuracy toward 99%. We expect that such a platform will contribute to expanding the toolbox for drug screening and characterization and facilitate the drug development process.

Dimension Reduction of Multidimensional Structured and Unstructured Datasets through Ensemble Learning of Neural Embeddings

Dimension reduction aims to project a high‐dimensional dataset into a low‐dimensional space. It tries to preserve the topological relationships among the original data points and/or induce clusters. NetDRm, an online dimensionality reduction method based on neural ensemble learning that integrates different dimension reduction methods in a synergistic way, is introduced. NetDRm is designed for datasets of multidimensional points that can be either structured (e.g., images) or unstructured (e.g., point clouds, tabular data). It starts by training a collection of deep residual encoders that learn the embeddings induced by multiple dimension reduction methods applied to the input dataset. Subsequently, a dense neural network integrates the generated encoders by emphasizing topological preservation or cluster induction. Experiments conducted on widely used multidimensional datasets (point‐cloud manifolds, image datasets, tabular record datasets) show that the proposed method yields better results in terms of topological preservation ( curves), cluster induction (V measure), and classification accuracy than the most relevant dimension reduction methods.

Abstract Mo030: Deep learning based multi-omics model for prediction of outcomes in HFpEF and HFmrEF

Introduction: Heart failure with preserved ejection fraction (HFpEF) presents a unique challenge in cardiology due to its diverse etiology and clinical presentation. Our study introduces a cutting-edge integrated deep learning model that combines multi-dimensional datasets, encompassing genetic variants from GWAS, cardiac structural and functional assessments from MRI, and electrophysiological data from ECGs, along with comorbidity profiles including CKD and LFT. Research Questions: Can a deep learning model that integrates multi-dimensional datasets accurately predict HFpEF risk? How does the inclusion of different types of data (genetic, MRI, comorbidity) affect the model's predictive accuracy? Goals: To develop a deep learning model that accurately predicts HFpEF risk To assess the impact of different types of data on the model's predictive accuracy. Methods: The model employs a graph convolutional multi-model deep learning approach to dissect the intricate interplay between genetic variants from GWAS, cardiac structural and functional assessments from MRI, electrophysiological data from ECGs, and comorbidity profiles including CKD and LFT. Results: The baseline genetic model achieved a PRS of 2.04 and an outcome accuracy of 81%. The incorporation of MRI features improved the outcome accuracy to 85.91%. The inclusion of comorbidities such as CKD and T2D resulted in the highest outcome accuracy of 88.57%. Discussion: The results suggest that the inclusion of MRI features and comorbidities improves the model's predictive accuracy. This model holds significant promise for clinical adoption and the advancement of personalized treatment strategies.

MKELM based multi-classification model for foreign accent identification

The automatic identification of foreign accents can play a crucial role in various speech systems, including speaker identification, e-learning, telephone banking, and more. Additionally, it can greatly enhance the robustness of Automatic Speech Recognition (ASR) systems. Non-native accents in speech signals are characterized by distinct pronunciations, prosody, and voice characteristics of the speaker. However, automatically identifying foreign accents poses significant challenges, particularly in the context of multi-class modeling. Multi-classification models face difficulties in achieving high performance and dealing with computational challenges when confronted with multi-dimensional and unbalanced datasets, such as those with more than two accents. Furthermore, the choice of features remains a bottleneck problem for Foreign Accent Identification (FAID), further hindering performance in these tasks. Consequently, the accuracy of current systems is typically low. To address these challenges, this paper proposes a framework based on the Multi-Kernel Extreme Learning Machine (MKELM) model for the multi-classification of FAID. The MKELM model utilizes a novel weighted scheme to classify various non-native English accents, including Arabic, Chinese, Korean, French, and Spanish. The model first combines Mel-frequency cepstral coefficients (MFCCs) and prosodic features as input, trains pairwise binary classifiers independently, and subsequently employs a weighting scheme to distinguish between classes and identify accents. Through experiments, the proposed model achieves an accuracy rate of 84.72% using a paired weighting scheme. In contrast, the accuracy rate drops to 66.5% when employing the traditional non-weighted multi-classification scheme. A comparison with other models demonstrates the significant advantages of the proposed model in FAID multi-class classification, showcasing improved accuracy, reduced computational complexity (requiring fewer computations, faster learning rates, and shorter training time), and enhanced stability compared to state-of-the-art classification methods.

A Grid-Based Method for Removing Overlaps of Dimensionality Reduction Scatterplot Layouts.

Dimensionality Reduction (DR) scatterplot layouts have become a ubiquitous visualization tool for analyzing multidimensional datasets. Despite their popularity, such scatterplots suffer from occlusion, especially when informative glyphs are used to represent data instances, potentially obfuscating critical information for the analysis under execution. Different strategies have been devised to address this issue, either producing overlap-free layouts that lack the powerful capabilities of contemporary DR techniques in uncovering interesting data patterns or eliminating overlaps as a post-processing strategy. Despite the good results of post-processing techniques, most of the best methods typically expand or distort the scatterplot area, thus reducing glyphs' size (sometimes) to unreadable dimensions, defeating the purpose of removing overlaps. This article presents Distance Grid (DGrid), a novel post-processing strategy to remove overlaps from DR layouts that faithfully preserves the original layout's characteristics and bounds the minimum glyph sizes. We show that DGrid surpasses the state-of-the-art in overlap removal (through an extensive comparative evaluation considering multiple different metrics) while also being one of the fastest techniques, especially for large datasets. A user study with 51 participants also shows that DGrid is consistently ranked among the top techniques for preserving the original scatterplots' visual characteristics and the aesthetics of the final results.

Bayesian Dictionary Learning on Robust Tubal Transformed Tensor Factorization.

The recent study on tensor singular value decomposition (t-SVD) that performs the Fourier transform on the tubes of a third-order tensor has gained promising performance on multidimensional data recovery problems. However, such a fixed transformation, e.g., discrete Fourier transform and discrete cosine transform, lacks being self-adapted to the change of different datasets, and thus, it is not flexible enough to exploit the low-rank and sparse property of the variety of multidimensional datasets. In this article, we consider a tube as an atom of a third-order tensor and construct a data-driven learning dictionary from the observed noisy data along the tubes of the given tensor. Then, a Bayesian dictionary learning (DL) model with tensor tubal transformed factorization, aiming to identify the underlying low-tubal-rank structure of the tensor effectively via the data-adaptive dictionary, is developed to solve the tensor robust principal component analysis (TRPCA) problem. With the defined pagewise tensor operators, a variational Bayesian DL algorithm is established and updates the posterior distributions instantaneously along the third dimension to solve the TPRCA. Extensive experiments on real-world applications, such as color image and hyperspectral image denoising and background/foreground separation problems, demonstrate both effectiveness and efficiency of the proposed approach in terms of various standard metrics.

Predicting forest fire probability in Similipal Biosphere Reserve (India) using Sentinel-2 MSI data and machine learning

The global escalation in forest fires, characterized by increasing frequency and severity, results from a complex interplay of natural and anthropogenic factors, exacerbated by climate change. These fires devastate habitats, threaten species, reduce biodiversity, disrupt natural cycles, and harm local ecosystems. The impacts are particularly damaging in biological reserves. The Similipal Biosphere Reserve (SBR) in Odisha State is one of India’s major forest fire hotspots, experiencing forest fires almost every year. The objective of this study is to develop a predictive model using Sentinel-2 MSI data and machine learning (ML) techniques to estimate the probability of forest fires in the SBR, India, thereby enhancing disaster management and prevention in the region. This research maps and quantifies forest fire intensity by leveraging ML algorithms, namely Extreme Gradient Boosting (XGBoost), Gradient Boosting Machine (GBM), Support Vector Machine (SVM), and Random Forest (RF). To develop a Forest Fire Probability (FFP) map, twenty conditioning factors, along with pre- and post-fire Normalized Burn Ratio (NBR) and delta Normalized Burn Ratio (dNBR), were utilized. Furthermore, four statistical methods—Mean Absolute Error, Mean Square Error, Root Mean Square Error, and Overall Accuracy—were employed to analyze the FFP. The results were validated using the Area Under Curve (AUC) method. The analysis identifies 2021 as the year with the highest incidence of forest fires, accounting for 29.19% of the occurrences. Among the models, the GBM exhibits superior performance, highlighting its efficacy in handling large, multidimensional datasets. Predictive mapping suggests that approximately 1400–1500 km2, or 25–30% of the studied area, faces a high to very high risk of forest fires.

ProteinFlow: An advanced framework for feature engineering in protein data analysis.

In the burgeoning field of proteins, the effective analysis of intricate protein data remains a formidable challenge, necessitating advanced computational tools for data processing, feature extraction, and interpretation. This study introduces ProteinFlow, an innovative framework designed to revolutionize feature engineering in protein data analysis. ProteinFlow stands out by offering enhanced efficiency in data collection and preprocessing, along with advanced capabilities in feature extraction, directly addressing the complexities inherent in multidimensional protein data sets. Through a comparative analysis, ProteinFlow demonstrated a significant improvement over traditional methods, notably reducing data preprocessing time and expanding the scope of biologically significant features identified. The framework's parallel data processing strategy and advanced algorithms ensure not only rapid data handling but also the extraction of comprehensive, meaningful insights from protein sequences, structures, and interactions. Furthermore, ProteinFlow exhibits remarkable scalability, adeptly managing large-scale data sets without compromising performance, a crucial attribute in the era of big data.

Naive Bayes classifier – An ensemble procedure for recall and precision enrichment

Data is essential for an organization to develop and make decisions efficiently and effectively. Machine learning classification algorithms are used to categorize observations into classes. The Naive Bayes (NB) classifier is a classification algorithm based on the Bayes theorem and the assumption that all predictors are independent of one another. Since this algorithm is based on probabilities, it is necessary to explore the sample distribution and feature type. This study presents an NB classifier method with enhanced performance among multidimensional and multivariate datasets, named the Naive Bayes Enrichment Method (NBEM). The NBEM is based on automated feature selection using threshold learning and the division of a dataset into sub-datasets according to the feature type. The main advantage of this method is the use of multiple NB classifiers based on different distributions and their combinations to classify a new observation. The final phase includes a weighted classification function that combines the results into a single output. This method was tested with 20 multivariate datasets and compared to other classification models and NB classifier variations. The results showed up to 76.3% improvement in the recall measure using NBEM and up to 43.9% improvement in the F1 score. Furthermore, we found that the error percentage of our method depended on the number of classes.

A comparative study of heterogeneous machine learning algorithms for arrhythmia classification using feature selection technique and multi-dimensional datasets

Arrhythmia, a common cardiovascular disorder, refers to the abnormal electrical activity within the heart, leading to irregular heart rhythms. This condition affects millions of people worldwide, with severe implications on cardiac function and overall health. Arrhythmias can strike anyone at any age which is a significant cause of morbidity and mortality on a global scale. About 80% of deaths related to heart disease are caused by ventricular arrhythmias. This research investigated the application of an optimized multi-objectives supervised Machine Learning (ML) models for early arrhythmia diagnosis. The authors evaluated the model’s performance on the arrhythmia dataset from the UCI ML repository with varying train-test splits (70:30, 80:20, and 90:10). Standard preprocessing techniques such as handling missing values, formatting, balancing, and directory analysis were applied along with Pearson correlation for feature selection, all aimed at enhancing model performance. The proposed optimized RF model achieved impressive performance metrics, including accuracy (95.24%), precision (100%), sensitivity (89.47%), and specificity (100%). Furthermore, the study compared the proposed approach to existing models, demonstrating significant improvements across various performance measures.

Comprehensive application of AI algorithms with TCR NGS data for glioma diagnosis

T-cell receptor (TCR) detection can examine the extent of T-cell immune responses. Therefore, the article analyzed characteristic data of glioma obtained by DNA-based TCR high-throughput sequencing, to predict the disease with fewer biomarkers and higher accuracy. We downloaded data online and obtained six TCR-related diversity indices to establish a multidimensional classification system. By comparing actual presence of the 602 correlated sequences, we obtained two-dimensional and multidimensional datasets. Multiple classification methods were utilized for both datasets with the classification accuracy of multidimensional data slightly less to two-dimensional datasets. This study reduced the TCR β sequences through feature selection methods like RFECV (Recursive Feature Elimination with Cross-Validation). Consequently, using only the presence of these three sequences, the classification AUC value of 96.67% can be achieved. The combination of the three correlated TCR clones obtained at a source data threshold of 0.1 is: CASSLGGNTEAFF_TRBV12_TRBJ1-1, CASSYSDTGELFF_TRBV6_TRBJ2-2, and CASSLTGNTEAFF_TRBV12_TRBJ1-1. At 0.001, the combination is: CASSLGETQYF_TRBV12_TRBJ2-5, CASSLGGNQPQHF_TRBV12_TRBJ1-5, and CASSLSGNTIYF_TRBV12_TRBJ1-3. This method can serve as a potential diagnostic and therapeutic tool, facilitating diagnosis and treatment of glioma and other cancers.

Leveraging Artificial Neural Networks for Robust Landslide Susceptibility Mapping: A Geospatial Modeling Approach in the Ecologically Sensitive Nilgiri District, Tamil Nadu

Landslides pose a significant threat to the lives and livelihoods of marginalised communities residing in rural areas and the delicate ecological balance of the environment. Implementing advanced technologies is crucial for improving hazard risk assessment and enhancing preparedness measures in regions characterised by diverse topography and complex geological formations. Geospatial applications and modelling techniques have emerged as indispensable in mitigating landslide risks, particularly in environmentally sensitive areas. This study presents a comprehensive approach to landslide susceptibility mapping in the Nilgiri District of Tamil Nadu, India, leveraging the power of Artificial Neural Networks (ANNs) and integrating multi-dimensional geospatial datasets. Integrating ANN-based modelling and geospatial techniques offers significant advantages in terms of statistical robustness, reproducibility, and the ability to analyse the complex interplay of factors influencing landslide hazards quantitatively. The methodology involves rigorous pre-processing and integrating spatial data, including landslide event occurrences as the dependent variable and ten independent parameters influencing landslide susceptibility. These parameters encompass elevation, slope aspect, slope degree, distance to roads, land use patterns, geomorphology, lithology, drainage density, lineament density, and rainfall distribution. Feature extraction and selection techniques are employed to effectively model the complex interactions between these factors and landslide occurrences. This process identifies the most relevant variables influencing landslide susceptibility, enhancing the model's predictive capabilities. The state-of-the-art ANNs are trained using historical landslide occurrence data and the selected influencing factors, enabling the development of a robust and accurate landslide susceptibility model. The performance of the developed model is rigorously evaluated using a comprehensive suite of metrics, including accuracy, precision, and the Area under the Receiver Operating Characteristic (ROC) curve. Preliminary results indicate that the ANN-based landslide susceptibility model outperforms traditional zonation methods, demonstrating higher accuracy and reliability in predicting landslide-prone areas. The resulting Landslide Susceptibility Map (LSM) categorises the study area into five distinct hazard zones, ranging from very high (664.1 km2), high (598.9 km2), moderate (639.7 km2), low (478.9 km2) and to very low (170.9 km2) . Notably, the eastern and central regions of the district emerge as particularly vulnerable to landslide occurrences. The study's findings have far-reaching implications for disaster risk reduction efforts, land-use planning, and sustainable development strategies in the ecologically sensitive Nilgiri District and beyond.

Frequent Pattern Associative Rules in Multiple Dimensions

This paper explores frequent pattern mining in multidimensional data and its extension beyond traditional one-dimensional approaches. While association rule mining is widely used in market basket analysis, its application to multidimensional datasets presents new challenges due to the increased complexity and sparsity of the data. This research reviews existing multidimensional mining techniques, focusing on frequent pattern associative rules, and evaluates their performance. Several real-world applications, such as business intelligence, healthcare analytics, and sensor networks, are discussed. The paper also proposes a framework for efficient pattern discovery and suggests areas for future research

Examination of Mandible in Morkaraman Sheep Using Geometric Morphometry Method

The aim of this study is to examine and analyze the mandible of Morkaraman sheep without any skeletal disorders using the geometric morphometric method. For this purpose, a total of 14 mandibles from male and female individuals were used in the study. The main components in multidimensional data sets were determined with Principal Component Analysis, used within the scope of the geometric morphometry method, and the differences between the samples were determined with Discriminant Function Analysis. Statistical and formal analyzes of these variances were also performed. A total of 12 principal components were obtained with 12 punctuations selected on a total of 14 mandibular (7 female, 7 male) images. Among these principal components, the first principal component (PC1) alone accounted for 30.409% of the total variation. The second principal component (PC2) alone accounted for 22.265% of the total variation, and the third principal component (PC3) alone accounted for 14.893% of the total variation. The first three of the variances obtained explained 67,567 of the shape differences. Discriminant Function Analysis (DFA) was used to objectively evaluate gender differences. In the discriminant function analysis, the p value was above 0.05 (p = 0.7). Although there was a complete separation between genders formally and statistically, no significant p value was obtained. In line with these analyses, information was obtained about the anatomical features and adaptations of the Morkaraman sheep mandible, which is one of the important economic resources of our country and is bred in a wide area, and it has become an exemplary study in this field.

Interactive tools for explaining multidimensional projections for high-dimensional tabular data

We present a set of interactive visual analysis techniques aiming at explaining data patterns in multidimensional projections. Our novel techniques include a global value-based encoding that highlights point groups having outlier values in any dimension as well as several local tools that provide details on the statistics of all dimensions for a user-selected projection area. Our techniques generically apply to any projection algorithm and scale computationally well to hundreds of thousands of points and hundreds of dimensions. We describe a user study that shows that our visual tools can be quickly learned and applied by users to obtain non-trivial insights in real-world multidimensional datasets. We also show how our techniques can help understanding a real-world dataset containing quantitative, ordinal, and categorical attributes.

Educational Big Data Analytics for Futuristic Smart Learning Using Deep Learning Techniques

The goal is to use the massive amounts of data created by digital education systems to develop intelligent and adaptable learning environments that are particularly suited to each student’s requirements. The rapid digitization of education systems has led to the proliferation of educational big data, presenting unprecedented opportunities to reshape learning environments into intelligent, responsive spaces that adapt to the needs of individual learners. This paper explores the integration of advanced deep learning techniques with educational big data analytics to forge the path towards futuristic smart learning ecosystems. By leveraging robust datasets derived from a myriad of educational interactions, ranging from student performance metrics to engagement patterns in digital learning platforms, we propose a multi-tiered analytical framework that harnesses the predictive power of deep learning. We commence by elucidating the scope and scale of educational big data, highlighting its potential to provide granular insights into student learning processes. The paper then delineates the architecture of a deep learning-based analytical model designed to process complex, multidimensional educational datasets. This model applies state-of-the-art algorithms to perform tasks such as predictive analytics for student performance, personalized content recommendation, and real-time engagement monitoring. Central to our discussion is the application of convolutional neural networks (CNNs), recurrent neural networks (RNNs), and deep belief networks (DBNs) in deciphering patterns and trends that escape traditional analytical methodologies. We emphasize the capacity of these techniques to capture the subtleties of learner behavior and to facilitate the development of adaptive learning pathways. Furthermore, we address the challenges of integrating deep learning with educational big data, including issues of data privacy, computational demands, and the need for robust model interpretation. The paper presents a series of case studies that demonstrate the successful application of our proposed framework in various educational settings, from K-12 to higher education and continuous professional development.

Synthetic Immunology—Building Immunity from the Bottom‐Up with Synthetic Cells

Synthetic cells can advance immunotherapy, offering innovative approaches to understanding and enhancing immune responses. This review article delves into the advancements and potential of synthetic cell technologies in immunology, emphasizing their role in understanding and manipulating immune functions. Recent progress in understanding vertebrate immune systems and the challenges posed by diseases highlight the need for innovative research methods, complementing the analysis of multidimensional datasets and genetic engineering. Synthetic immune cell engineering aims to simplify the complexity of immunological systems by reconstructing them in a controlled setting. This approach, alongside high‐throughput strategies, facilitates systematic investigations into immunity and the development of novel treatments. The article reviews synthetic cell technologies, focusing on their alignment with the three laws of immunity: universality, tolerance, and appropriateness. It explores the integration of synthetic cell modules to mimic processes such as controlled T‐cell activation, bacteria engulfment and elimination, or cellular maturation into desirable phenotypes. Together, such advancements expand the toolbox for understanding and manipulating immune functions. Synthetic cell technologies stand at the innovation crossroads in immunology, promising to illuminate fundamental immune system principles and open new avenues for research and therapy.

Multi-dimensional classification via class space fusion and comprehensive label correlations

Multi-dimensional Classification (MDC) is a new multi-output learning paradigm, where each instance in this framework is annotated with labels from multiple semantic class spaces. The vital challenge in MDC problem is how to address the heterogeneity of output space. Existing methods typically focus on the decomposition of output space, rarely consider the comprehensive fine-grained label correlations and the exploration of the intrinsic structure of the output space. And the local label correlations, which only exist within specific groups of instances, remain largely untapped in existing MDC methods. Therefore, we focus on tackling these above problems in this paper by simultaneously conducting label space fusion and considering the comprehensive label correlations to fully exploit the latent discriminative information hidden in the output space. Heuristically, we utilize low-rank representation method to achieve label space fusion to capture the intrinsic structure and implicit global correlation of label space. Then the predictive MDC model is extrapolated from the reconstructed instances equipped with the fused label space. Moreover, the topological structure information of samples is fully taken advantage of for MDC model establishment. Benefiting from these strategies, the heterogeneous label space can be fully excavated to boost the performance of MDC. Empirically, comparative experiments are conducted over several multi-dimensional benchmark data sets, and corresponding experimental results firmly illustrate that the proposed algorithm can effectively solve the MDC problem and outperform state-of-the-art methods.

Generalized sparse and outlier-robust broad learning systems for multi-dimensional output problems

Broad learning systems (BLSs) are becoming increasingly popular due to their fast and superior learning capabilities. However, their performances are susceptible to outliers and dense networks. The Elastic-net robust BLS (ER-BLS), which employs an ℓ1-norm loss function along with Elastic-net regularization, attempts to mitigate these issues by promoting network sparsity. Nevertheless, even ER-BLS fails to achieve satisfactory robustness and sparsity in multi-dimensional output tasks because the ℓ1-norm ignores the correlations between the output dimensions and output weights. Therefore, we propose a generalized Elastic-net robust BLS (GER-BLS) by generalizing the ℓ1-norm to the L2,1-norm. The L2,1-norm enhances robustness by treating the residuals across all output dimensions as a whole and induces node-level sparsity by penalizing all output weights connected to a single hidden node. The alternating direction method of multipliers (ADMM) algorithm was employed to solve GER-BLS efficiently, circumventing the non-differentiable issue of the objective function. Moreover, we developed DGER-BLS, a distributed variant of GER-BLS, tailored to handle data with distributed storage. The convergence analysis of our proposed algorithms and a large number of numerical experiments on multi-dimensional output datasets demonstrated the effectiveness of GER-BLS and DGER-BLS in dealing with outliers, distributed stored data, and sparsity.