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RoBell-RVFL: A Robust Generalized Bell Random Vector Functional Link Network

A. Rahaman, A. Quadir, M. Tanveer

arXiv:2608.16965v1cs.LG

TL;DR

Imbalanced, noisy data can bias RVFL decision boundaries because existing methods overlook the quality of majority-class samples while preserving minority information. RoBell-RVFL uses probability-weighted generalized-bell sample weighting and consistently outperforms competing RVFL variants across benchmark datasets and label-noise conditions.

  • Problem

    Existing imbalance methods address class quantity but lack mechanisms to suppress noisy majority samples while preserving discriminative minority information.

  • Method

    RoBell-RVFL adaptively weights majority samples with kernel-induced, probability-weighted gbell memberships while assigning minority samples unit weights.

  • Results

    81.8927% average accuracy across 26 benchmark datasets surpassed NF-RVFL’s 80.5156%, while performance remained robust under increasing label noise.

  • Takeaways & Limitations

    The results support jointly considering sample quality, class dominance, and noise suppression when learning from imbalanced data.

Abstract

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The dominance of majority classes in real-world datasets poses a fundamental challenge to randomized neural networks, often biasing decision boundaries and overlooking critical minority samples. Existing remedies, such as synthetic minority over-sampling (SMOTE) and class-weighted loss functions, primarily address class proportions while neglecting intra-class distribution, making them vulnerable to label noise and outliers. In this paper, we propose \textbf{RoBell-RVFL}, a robust and lightweight \emph{quality-aware} generalized bell random vector functional link network that redefines how randomized models handle class imbalance and noisy data. RoBell-RVFL employs a dual-strategy, sample-level weighting mechanism that strictly preserves minority class information using unit weights, while adaptively regulating the influence of majority class samples through a probability-weighted generalized bell (gbell) membership function in a kernel-induced feature space. This design effectively suppresses noisy, boundary, and outlier samples within the majority class, enabling the network to learn from informative samples rather than merely abundant ones. By explicitly incorporating local class probability and class distribution information into the learning process, RoBell-RVFL achieves adaptive control over sample contributions without sacrificing the closed-form learning efficiency of RVFL networks. Extensive evaluations on UCI and KEEL benchmark datasets, along with robustness tests under up to 40\% label noise, demonstrate that RoBell-RVFL consistently and significantly outperforms recent state-of-the-art RVFL variants. The results indicate that adaptive, quality-aware sample weighting is essential for robust RVFL learning, rendering conventional global weighting schemes ineffective in noisy and imbalanced environments.

I. INTRODUCTION AND MOTIVATION

RVFL offers fast, low-complexity learning through a shallow architecture and closed-form output-weight estimation, but imbalanced data with majority-class noise and outliers can bias decision boundaries. RoBell-RVFL addresses this gap with sample-level, probability-weighted generalized bell membership that preserves minority information while suppressing uninformative majority samples.

  • RVFL motivation: RVFL combines a shallow single-hidden-layer architecture, direct input-output links, and closed-form output-weight learning to achieve rapid training and low computational complexity.Its randomized hidden-layer weights avoid iterative updates and gradient-based optimization.
  • Problem motivation: Severely imbalanced real-world datasets often contain majority-class noise and outliers that overwhelm minority samples and distort classification.The paper identifies this challenge in applications including face recognition and medical diagnostics.
  • Limitations of existing methods: SMOTE and related rebalancing methods address class proportions but may discard informative majority samples, introduce redundancy, or amplify noise.The paper distinguishes data-level undersampling and oversampling approaches from algorithm-level modifications.
  • Limitations of existing methods: Existing methods largely overlook sample quality, lacking mechanisms to suppress noisy or outlier majority samples while preserving minority discriminative information.Consequently, majority-class noise continues to bias decision boundaries in imbalanced learning scenarios.
  • RoBell-RVFL contribution: RoBell-RVFL uses dual-strategy sample-level weighting: minority samples receive unit weights, while majority-sample influence is adaptively reduced according to distribution characteristics.This fine-grained weighting replaces coarse class-level reweighting.
  • RoBell-RVFL contribution: A probability-guided generalized bell membership function combines spatial distribution information with local class probability to distinguish informative samples from majority-class noise and outliers.The mechanism is designed to prevent majority-class dominance during learning.

II. NOTATION

This section defines the training-data notation for inputs, one-hot targets, dataset dimensions, and the matrices that collect samples and labels.

  • Dataset notation: The training set contains A samples (a_i, b_i), with a_i ∈ R^(1×B) and b_i ∈ R^(1×D), where B is features and D is classes.Here, A denotes the total number of training samples.
  • Dataset notation: Each target vector b_i is one-hot encoded to represent the class label of input sample a_i.
  • Matrix notation: The matrices Z and Z_out collect all input samples and corresponding target values, respectively, with (·)^t denoting transpose.D also denotes the number of hidden nodes in this notation.

III. PROPOSED METHOD

The proposed framework distinguishes majority and minority samples while addressing imbalance through adaptive weighting, class-dependent regularization, and kernel-induced feature mapping.

  • Sample representation: Training samples are represented by B-dimensional feature vectors with binary labels, and the framework distinguishes majority from minority class samples.Positive and negative class samples are organized into separate matrices before method application.
  • Adaptive weighting: The method avoids explicit imbalance-ratio use when defining sample weights.Class imbalance is instead handled through adaptive sample weighting and class-dependent regularization during learning.
  • Kernel-induced representation: Kernel-induced mapping implicitly projects samples into a higher-dimensional feature space for modeling nonlinear class distributions.The mapping is denoted by K(·).

A. Determination of Membership Value

RoBell-RVFL uses a generalized bell membership function to assign adaptive sample weights in a kernel-induced feature space. It suppresses majority-class outliers and boundary samples while preserving full membership for minority samples.

  • Membership assignment: The gbell function assigns adaptive membership values according to sample distributions in a kernel-induced feature space.Its smooth controllability supports flexible fuzzy membership modeling.
  • Class-aware weighting: Majority-class outliers and boundary samples are down-weighted, while minority-class samples retain full membership to avoid information loss.This weighting design regulates majority influence without discarding minority information.
  • Kernel scale: The kernel width parameter ξ controls Gaussian-kernel similarity scale: smaller ξ emphasizes local neighborhoods, whereas larger ξ produces smoother similarity distributions.Thus, ξ determines the spatial scale used in the kernel-induced feature space.

1) Determination of Class Probability:

RoBell-RVFL assigns each training sample a class probability that measures local class consistency in a kernel-induced neighborhood. These probabilities suppress noisy, mislabeled, boundary, and outlier samples by reducing their training influence, particularly in the majority class.

  • Determination of Class Probability: Each training sample receives a class probability reflecting its local class consistency within a kernel-induced neighborhood.The probability is used to reduce the influence of noise and ambiguous samples during training.
  • Determination of Class Probability: Unlike fixed-radius input-space methods, RoBell-RVFL computes neighborhood relations using a kernel-based distance measure.The local neighborhood is defined by samples whose kernel-induced distance satisfies d(a_i, a_j) ≤ α_d.
  • Determination of Class Probability: The class probability equals the ratio of same-class samples within each sample’s local neighborhood, with ε preventing numerical instability.The neighborhood is N(a_i) = {a_j | d(a_i, a_j) ≤ α_d}.
  • Determination of Class Probability: Higher p(a_i) indicates stronger local class consistency, whereas lower values identify likely noisy or boundary samples assigned lower training influence.This adaptive weighting improves RoBell-RVFL robustness, especially for majority-class samples.

2) PW-GB Membership Function:

The PW-GB membership function combines generalized bell membership values with local class probability to weight samples in a kernel-induced feature space. It suppresses noisy and boundary majority samples while preserving minority-sample contributions.

  • PW-GB Membership Function: PW-GB combines generalized bell membership values with local class probability to construct a probability-weighted membership function.The function is designed for sample weighting in the proposed method.
  • PW-GB Membership Function: The function suppresses noisy and boundary samples from the majority class while preserving the full contribution of minority samples.This weighting strategy targets majority-class sample quality without reducing minority-class information.
  • PW-GB Membership Function: ξpw(ai) is the gbell membership value computed in kernel-induced feature space, and p(ai) is the sample’s local class probability.These quantities define the membership function’s feature-space and probability components.

B. Class Imbalance Weighting Scheme

The weighting scheme preserves minority-class information by assigning minority samples unit weight, while reducing majority-sample influence according to the minority-to-majority class-size ratio.

  • Minority-class samples are assigned a weight of one.
  • Majority-class samples are down-weighted proportionally to the ratio of minority to majority instances.
  • The strategy contrasts fixed unit weighting for minority samples with ratio-based weighting for majority samples.

IV. PROPOSED ROBELL-RVFL

RoBell-RVFL formulates class-imbalance-aware learning through sample-weighting matrices that regulate class-specific contributions. Its optimization uses nonlinear and linear projections, hidden representations, and a compact matrix solution for the output-layer weights.

  • Class-imbalance-aware formulation: RoBell-RVFL integrates a class-imbalance-aware weighting mechanism that adjusts sample contributions according to the imbalance ratio.The formulation is designed to mitigate adverse effects of imbalanced data.
  • Class-imbalance-aware formulation: L− and L+ are diagonal matrices containing PW-GB membership values for negative and positive class samples, respectively.These matrices encode class-specific membership values in the optimization formulation.
  • Optimization formulation: The optimization uses X− and X+ as nonlinear and linear projections of negative and positive class samples, with γ ∈ R+ penalizing error variables.β− and β+ denote class-specific error vectors, while l− and l+ define the corresponding weighting schemes.
  • Hidden-layer representations: Z− and Z+ are projected through randomly initialized weights and biases, followed by nonlinear activation ϕ, to produce hidden outputs Y− and Y+ that also serve as target outputs.These matrices represent the class-specific input data and hidden-layer outputs used in the model.
  • Class-specific weighting and solution: CI-aware weighting matrices l+ and l− regulate the influence of majority and minority class samples.The formulation subsequently derives a compact matrix form used to obtain the output-layer weight matrix.

A. Evaluation Dataset

Across 26 benchmark datasets, RoBell-RVFL achieves the highest average accuracy and lowest average rank among the evaluated models. Statistical testing further confirms significant performance differences across the compared methods.

  • Classification accuracy: 81.8927% average classification accuracy makes RoBell-RVFL the top method across 26 benchmark datasets, ahead of NF-RVFL at 80.5156%.Detailed per-dataset results, standard deviations, and optimal hyperparameters are provided in Supplementary Table IV.
  • Classification accuracy: 79.2817% and 78.8248% average accuracies are achieved by RVFL and RVFLwoDL, while CRVFL, ACRVFL, GE-GB-RVFL, and GB-RVFL obtain 76.5716%, 78.5455%, 79.6417%, and 79.8155%, respectively.These conventional, complex-valued, and granular ball-based RVFL variants all trail the proposed method in average accuracy.
  • Average rank: 3.2308 lowest average rank is obtained by RoBell-RVFL, followed by GE-GB-RVFL at 4.0000 and GB-RVFL at 4.0577.NF-RVFL, CRVFL, and ACRVFL have higher average ranks of 4.2692, 5.7308, and 4.5962, respectively.
  • Statistical significance: 2.8202 exceeds the 5% critical F-distribution value of 2.0622, rejecting equal average ranks across eight models evaluated on 26 datasets.The corresponding Friedman statistic is χ2_F = 18.4495, with the critical value based on (7, 175) degrees of freedom.

V. CONCLUSION AND FUTURE WORK … S.VII. EVALUATION OF ROBELL-RVFL UNDER LABEL NOISE

RoBell-RVFL combines adaptive probability-weighted generalized-bell membership with closed-form RVFL learning to address imbalance, noise, and outliers. Across benchmark and label-noise evaluations, it achieves robust performance and statistically supported gains over competing models.

  • V. CONCLUSION AND FUTURE WORK: RoBell-RVFL adaptively weights samples using kernel-induced gbell membership and local class probabilities, suppressing noisy majority samples while preserving minority information.Class imbalance is handled through class-dependent regularization rather than heuristic ratio-based weighting.
  • A. RVFL: RVFL uses randomly fixed input-to-hidden weights and biases, concatenates input and hidden features, and learns output weights through a pseudoinverse or least-squares solution.This architecture provides the closed-form learning foundation used by the proposed model.
  • S.II. COMPUTATIONAL COMPLEXITY: RoBell-RVFL training combines PW-GB membership computation with closed-form RVFL learning, with overall time complexity O(A^2B + ABD + A(B + D)^2 + (B + D)^3).The complexity includes kernel-based membership computation and solving for the output weight matrix.
  • S.III. ALGORITHM OF THE PROPOSED ROBELL-RVFL: The algorithm partitions majority and minority samples, computes Gaussian-kernel probabilities and PW-GB memberships, applies class-imbalance regularization, and obtains output weights in closed form.It constructs hidden outputs and concatenated features before solving for V.
  • S.IV. EXPERIMENTAL SETUP, COMPARED MODELS AND DATASETS: Experiments use a 70:30 train-test split with fivefold cross-validation and grid search on a workstation running Python 3.11.Comparisons include diverse randomized neural-network architectures and tuned activation functions.
  • Performance Metrics: Evaluation reports Accuracy, Sensitivity, Specificity, Precision, F-measure, and G-mean to assess predictive performance under class imbalance and label noise.F-measure and G-mean jointly reflect class-wise performance and balance between sensitivity and specificity.
  • S.V. THEORETICAL PROOF ON THE BOUND OF THE PW-GB MEMBERSHIP FUNCTION: The PW-GB membership function is bounded by 0 ≤ ξ(ai) ≤ 1 for all training samples.The proof considers minority and majority cases and uses kernel-induced radii together with local class probabilities.
  • S.VI. WIN–TIE–LOSS (W–T–L) SIGN TEST:; S.VII. EVALUATION OF ROBELL-RVFL UNDER LABEL NOISE: RoBell-RVFL records 18 wins against RVFLwoDL and CRVFL, 17 against RVFL and ACRVFL, 16 against GB-RVFL, and 15 against NF-RVFL and GE-GB-RVFL.With ω = 26, the W–T–L test uses 18 wins as the significance threshold; under label noise, average accuracy is 78.4137% versus 76.7610% for RVFL and 76.4659% for GB-RVFL.

S.VIII. SENSITIVITY ANALYSIS

The sensitivity analysis evaluates how the regularization parameter ξ and parameter ρ affect RoBell-RVFL accuracy individually and jointly across three representative datasets. Results show generally stable performance, but effective tuning ranges remain dataset-specific, especially for ξ.

  • Analysis scope: The analysis varies ξ and ρ separately and jointly to assess RoBell-RVFL’s classification accuracy, robustness, and stability on three representative datasets.The tested effects include ξ, ρ, and their joint influence on model performance.
  • Effect of ξ: For heart hungarian, accuracy improves as ξ increases to 4 before degrading, while vehicle2 declines when ξ > 2 and led7digit remains relatively stable.The led7digit-0-2-4-5-6-7-8-9 vs 1 3 dataset reaches its peak accuracy at an omitted value in the supplied passage.
  • Effect of ρ: Increasing ρ generally improves performance, with heart hungarian and led7digit-0-2-4-5-6-7-8-9 vs 1 3 reaching maximum accuracy when ρ ≥0.625.The passage reports gradual improvement for most datasets, including vehicle2, and indicates that higher ρ values generally promote stable and improved learning.
  • Joint effect of ξ and ρ: Joint sensitivity results show peak performance around ρ = 1 and ξ ≈0.5, indicating robustness to joint parameter variation.The passage states that once ξ ≥2, accuracy remains stable across most values of ρ, although the supplied text is truncated.
  • Dataset-specific tuning: Dataset-specific tuning ranges are identified: heart hungarian uses ρ close to 1 and ξ ∈[1, 32], whereas vehicle2 performs best with ξ ∈[0.125, 2] and ρ ≈ 1.For vehicle2, larger ξ values cause substantial degradation; the supplied passage truncates the corresponding led7digit range.
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