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IFW-BLS: Dual-Robust Broad Learning System with Intuitionistic Fuzzy Wave Loss
Mushir Akhtar, M. Tanveer
TL;DR
Standard BLS can be distorted by extreme residuals and by treating ambiguous samples as equally reliable. IFW-BLS combines intuitionistic-fuzzy credibility weighting with bounded, asymmetric wave-loss protection and evaluates the resulting objective using accelerated gradient optimization. Across UCI benchmarks and corruption tests, it reports stronger overall performance and greater stability than standard BLS, while remaining statistically competitive with closely related robust alternatives.
Problem
Standard BLS least-squares training is vulnerable to extreme errors and does not distinguish the reliability of ambiguous or locally conflicting samples.
Method
IFW-BLS weights residuals by intuitionistic-fuzzy credibility based on global and local information, then applies bounded asymmetric wave loss and Nesterov accelerated gradient optimization.
Results
IFW-BLS achieves the best average accuracy and average rank across 30 UCI datasets, significantly outperforming several baselines while remaining statistically competitive with Wave-RVFL and IF-BLS.
Takeaways & Limitations
Combining sample credibility with residual protection gives IFW-BLS stronger overall benchmark performance and more stable accuracy than standard BLS under noise and outlier contamination.
Abstract
from arXiv · showhide
Broad Learning System is an efficient randomized learning model that expands network width through feature and enhancement nodes and estimates the output weights without deep backpropagation. Its standard least-squares training, however, is vulnerable in two different ways: (i) large residuals caused by noise, outliers, or corrupted labels can dominate the objective, and (ii) all samples are treated as equally reliable even when some lie in ambiguous or locally conflicting regions. This paper proposes IFW-BLS, an Intuitionistic Fuzzy Wave Broad Learning System that addresses these two sources of fragility within one optimization model. The first robustness mechanism is residual-level protection, obtained by replacing the squared loss with the bounded, smooth, and asymmetric wave loss. Boundedness prevents extreme residuals from receiving unbounded influence, while asymmetry allows positive and negative deviations to be penalized differently when the dominant error direction varies. The second mechanism is sample-level credibility control, obtained through intuitionistic fuzzy scores that combine global class-center consistency with local neighborhood conflict. The resulting model evaluates the wave loss on credibility-weighted residuals, so unreliable samples are down-weighted before the bounded loss further limits the effect of extreme errors. A Nesterov accelerated gradient based optimizer is used to solve the proposed objective, avoiding the explicit matrix inversion used in conventional BLS. Experiments on UCI benchmark datasets validate the superiority of the proposed IFW-BLS model over the baseline models; additional corruption experiments also show more stable performance than BLS under noise and outlier contamination.
1 Introduction
IFW-BLS combines intuitionistic-fuzzy sample credibility with bounded wave-loss residual protection to address complementary weaknesses in standard BLS. The paper evaluates this unified approach on benchmark datasets and under controlled corruption.
- Motivation: Standard BLS is vulnerable because quadratic penalties let noisy observations, outliers, and mislabeled samples disproportionately affect learned output weights.Its regularized least-squares training is also limited by treating uncertain samples without differentiated reliability control.
- Proposed approach: IFW-BLS combines sample-level credibility modeling with residual-level robustness in one Broad Learning System.Intuitionistic fuzzy scores quantify training-sample credibility, while the wave loss protects against extreme residuals.
- Proposed approach: The wave loss is bounded, smooth, and asymmetric, limiting extreme deviations while allowing different penalties for positive and negative residuals.This design supports datasets where overestimation-like and underestimation-like deviations are not equally harmful.
- Proposed approach: Intuitionistic fuzzy scores use global class-center information and local class-conflict information to reduce the influence of ambiguous or unreliable samples.The credibility view controls how strongly a sample influences learning, complementing the residual view's error penalty.
- Optimization and evaluation: The credibility-weighted wave-loss objective softens unreliable samples before robust-loss evaluation and is optimized with a Nesterov accelerated gradient strategy.This avoids the explicit matrix inversion used in conventional BLS training.
- Optimization and evaluation: Experiments use 30 UCI datasets, accuracy and average rank, Friedman and Nemenyi tests, plus controlled noise and outlier contamination.The evaluation compares IFW-BLS with randomized, BLS, fuzzy, intuitionistic-fuzzy, wave-loss, and kernel-based baselines.
2 Preliminaries
BLS constructs a broad representation by concatenating randomized feature and enhancement nodes, while intuitionistic fuzzy credibility scores assess sample reliability and wave loss limits and reshapes residual penalties.
- Broad Learning System: BLS maps inputs through randomized feature windows, transforms them with enhancement windows, and concatenates the results into a broad representation for output-weight learning.
- Broad Learning System: The feature windows use randomly generated parameters and an activation function to produce feature-node outputs.
- Broad Learning System: The enhancement layer maps the concatenated feature nodes into multiple enhancement windows before forming the final representation.
- Broad Learning System: The conventional regularized least-squares BLS solution uses a matrix inverse, which becomes expensive for broad representations or large training sets and remains sensitive to extreme residuals.
- Wave Loss: Wave loss is bounded below 1/λ, with parameter a controlling whether positive or negative residuals receive stronger penalties.
- Intuitionistic Fuzzy Credibility: Intuitionistic fuzzy credibility combines class-center consistency with local class conflict to represent sample reliability.
3 Proposed Dual-Robust BLS
IFW-BLS combines credibility-weighted residuals with bounded, asymmetric wave loss, then optimizes the smooth objective using Nesterov accelerated gradient instead of conventional matrix inversion.
- Dual-Robust Objective: IFW-BLS scales each residual by sample credibility before applying wave loss, combining sample-level down-weighting with residual-level bounded protection.
- Dual-Robust Objective: The proposed objective uses regularization together with wave-loss terms evaluated on credibility-weighted residuals.
- Dual-Robust Objective: Zero-credibility samples contribute neither loss nor gradient, while intermediate credibility contracts residuals and adds a multiplicative factor to gradient contributions.
- Optimization: The objective is smooth for fixed credibility scores and randomized features, enabling optimization with a Nesterov accelerated gradient procedure.
- Optimization: NAG initializes the BLS representation, credibility scores, output weights, and velocity, then uses look-ahead evaluation, residual and gradient computation, and decayed learning-rate updates.
- Complexity: Over T iterations, optimization costs O(Tmnbc), while total training costs O(mdpq + mpqrs + m2d + Tmnbc), excluding lower-order terms.
- Complexity: Compared with conventional BLS inversion, IFW-BLS uses first-order updates and adds a one-time credibility computation.
4 Experiments and Discussion
IFW-BLS is evaluated against eight baselines on 30 UCI datasets using accuracy, rank, statistical tests, and controlled corruption experiments. It achieves the strongest overall clean-data performance and retains higher accuracy than standard BLS as feature outliers and label noise increase.
- Performance comparison: IFW-BLS achieves the best average accuracy, 87.4896%, across 30 UCI datasets, exceeding IF-BLS by 3.5794% and standard BLS by 5.4145%.The comparison includes RVFL, RVFLwoDL/ELM, BLS, Wave-RVFL, NF-BLS, F-BLS, IF-BLS, and KRP-BLS.
- Performance comparison: IFW-BLS records the best average rank of 2.3333 and the best or tied-best result on 21 of 30 datasets, with second-best results on 3 additional datasets.It wins or ties against IF-BLS on 24 datasets and against standard BLS on 27 datasets.
- Statistical significance: The Friedman analysis rejects equal model performance: the corrected statistic is F = 13.0014 versus the 5% critical value F(8, 232) = 1.9784.Nine methods are compared across 30 datasets.
- Statistical significance: Nemenyi testing finds IFW-BLS significantly better than RVFL, RVFLwoDL, BLS, NF-BLS, F-BLS, and KRP-BLS, but not significantly different from Wave-RVFL or IF-BLS.Wave-RVFL and IF-BLS are the closest robust alternatives because they separately use wave-loss robustness or intuitionistic fuzzy credibility.
- Robustness analysis: Under 5%–20% feature-outlier and label-noise contamination, IFW-BLS consistently preserves higher accuracy than standard BLS on blood and horse_colic.Across both datasets, average accuracy is 81.6346% under outliers and 82.7312% under noise for IFW-BLS, versus 75.8949% and 77.2617% for BLS.
5 Conclusions
IFW-BLS improves standard BLS robustness through complementary sample-level reliability modeling and residual-level protection in a unified objective.
- IFW-BLS measures each training sample’s credibility using intuitionistic fuzzy scores based on global class-center consistency and local neighborhood conflict.Ambiguous, noisy, or class-inconsistent samples receive reduced influence during training.
- The model replaces squared loss with a bounded, asymmetric wave loss that limits large-residual dominance and differentiates positive from negative deviations.The asymmetry allows penalties to vary with data characteristics.
- IFW-BLS combines both mechanisms by scaling residuals with intuitionistic fuzzy scores before evaluating them through the wave loss.This forms a unified credibility-weighted wave-loss objective.