HiER-BLS: A Hierarchy-Guided and Error-Correcting Robust Incremental Broad Learning System
Authors: Gongli Zhang, C. L. Philip Chen, Zhulin Liu
Organizations: Guangdong Provincial Key Laboratory of Computational AI Models and Cognitive Intelligence, the School of Computer Science and Engineering, South China University of Technology, Guangzhou 510006, China, also with the Pazhou Lab, Guangzhou 510335, China, and also with the Engineering Research Center of the Ministry of Education on Health Intelligent Perception and Paralleled Digital-Human, Guangzhou 510641, China
Broad Learning System (BLS) supports analytical training and incremental expansion, but its growth needs guidance on which inputs new blocks should learn from. Weight errors pose a further challenge by displacing learned outputs across class boundaries. We propose HiER-BLS to couple hierarchy-guided representation growth with error-correcting learning. Successive blocks focus on inputs selected by feature importance and correlation while preserving earlier representations. The evolving branch guides encoded learners through subspace size and sample confidence, so its learning experience informs both their feature views and supervision. For finite broad readouts, we show how codeword correlations transform fitted class scores. Prediction preservation depends on the distance from the actual output to the nearest decoding boundary relative to the model's sensitivity to weight errors. Experiments on five image and five tabular datasets demonstrate improved classification performance over representative BLS variants. Component studies show that hierarchy guidance benefits the encoded branch even when the guiding branch has lower standalone accuracy, with further gains from combining their scores. Longer codes continue to improve accuracy under stronger Gaussian weight errors after clean accuracy has largely saturated.
Figures & tables
Fig. 1 : Architecture of HiER-BLS. HGEE constructs incremental features using importance-guided subspaces. ECEO trains encoded bases and prunes output columns. BADF combines the branches’ scores through an additional BLS.
Importance profile
Schedule
Raw size st
Dense
Exponential
Mγt−1
Uniform
Linear
M[1−α(t−1)]
Mixed
Cosine
2M[1+cos(π(t−1)/T)]
TABLE I : Raw subspace schedules selected by the importance profile.
Operation
Leading cost
Spearman correlations and cached FSR
O(MNlogN+NM2+M2)
ECB b , qb=fb+hb
Cmap(M,fb,hb)+Cfit(qb,Lb,0)
PCP over all added columns
O(NvCL03) directly; O(NvCL02) with cached scores
BADF, qBA=fBA+hBA
Cmap(2C,fBA,hBA)+Cfit(qBA,C)
TABLE II : Costs outside the HGEE updates. Here fb,hb and fBA,hBA are the feature and enhancement widths of an ECB and BADF, respectively.
Dataset
Type
Input
Class
Train
Test
HGEE
ECEO
BADF
λ
Feature
Enh.
Feature
Enh.
L
Feature
Enh.
MNIST
Image
(28,28,1)
10
60000
10000
(30,40)
(1,11000)
(10,10)
(1,11000)
20
(20,10)
(1,30)
10−6
Fashion-MNIST
Image
(28,28,1)
10
60000
10000
(100,10)
(1,9000)
(100,10)
(1,11000)
20
(30,40)
(1,770)
10−6
NORB
Image
(32,32,2)
5
24300
24300
(10,100)
(1,8000)
(30,50)
(1,8000)
10
(50,35)
(1,10)
10−3
CIFAR10
Image
(32,32,3)
10
50000
10000
(128,20)
(1,5000)
(128,20)
(1,10000)
20
(20,25)
(1,22)
10−7
CIFAR100
Image
(32,32,3)
100
50000
10000
(128,20)
(1,5000)
(128,20)
(1,10000)
200
(10,10)
(1,10)
10−6
TABLE III : Dataset sizes and model settings. Feature and Enh. specify the feature- and enhancement-window configurations, respectively, as (number of windows, nodes per window).
TABLE IV : Image-classification accuracy (%). Avg and Std denote the reported mean and standard deviation. Bold marks the largest mean or smallest standard deviation in each column.
TABLE V : Testing accuracy and training time on tabular datasets.
Fig. 2 : Feature-importance distributions (left) and HGEE accuracy versus evolution step for selected schedule parameters (right).
Fig. 3 : Classification accuracy versus ECOC code length under zero-mean Gaussian weight errors. Each panel includes three fixed values of σ ; dashed horizontal lines denote the corresponding one-hot baselines.
TABLE VI : Testing accuracies (%) of HiER-BLS and its ablated variants. HTI, SA, ASR, SS, and CS denote hierarchy type identification, subspace alignment, AdaBoost sample reweighting, subspace sampling, and confidence scaling. Removing HTI means randomly selecting one of the other two schedules.
Fig. 4 : Two-dimensional t-SNE projections of HGEE features on CIFAR100 without and with subspace alignment. Colors identify the full feature space and subspaces 1–4.
Fig. 5 : CIFAR10 accuracy (%) and LTSC (%) across ensemble stages, with and without PCP. LTSC is displayed as 100× the value in Eq. ( 72 ).
Broad Learning System (BLS) offers an efficient alternative to deep architectures by enabling fast learning through randomized feature mapping and closed-form solutions. However, its reliance on squared error loss makes it highly sensitive to noise, outliers, and corrupted labels, limiting its reliability in real-world scenarios. To address this limitation, we propose Wave-BLS, a robust broad learning framework that integrates the wave loss function, which is asymmetric, bounded, and smooth, enabling controlled penalization of large errors. The proposed formulation replaces the standard least-squares objective with a wave-loss-based optimization problem, solved efficiently using a Nesterov accelerated gradient (NAG)-based scheme without requiring matrix inversion, thereby improving scalability. Extensive experiments on 30 UCI benchmark datasets demonstrate that Wave-BLS consistently outperforms classical BLS and several robust variants. Statistical validation using Friedman and Nemenyi post-hoc tests confirms the significance of the observed improvements. Furthermore, robustness evaluations under controlled noise and outlier injection reveal that Wave-BLS exhibits substantially slower performance degradation compared to BLS, even in challenging contamination settings. These results establish Wave-BLS as a stable and robust alternative to existing broad learning models for learning under data uncertainty.
Mushir Akhtar, A. Varshney, A. Quadir +3
Department of Mathematics Indian Institute of Technology Indore
Broad Learning System (BLS) is an efficient alternative to deep architectures due to its fast training, analytical learning, and strong generalization under limited data. However, existing BLS variants are confined to real-valued representations, restricting their ability to capture nonlinear interactions and second-order statistical dependencies inherent in real-world data. Notably, no prior BLS model fully exploits the complete second-order statistics that naturally emerge when data are embedded in the complex domain. To address this limitation, this paper introduces the first complex augmented Broad Learning System (CA-BLS), which transforms real-valued inputs into phase-encoded complex representations and adopts widely linear modeling to jointly leverage covariance and pseudo-covariance information via complex conjugate augmentation. This enables effective modeling of latent nonlinearities, coherence structures, and second-order dependencies inaccessible to conventional BLS formulations. To mitigate the additional computational cost of complex augmentation, an Efficient Complex Augmented BLS (ECA-BLS) is further developed, reformulating CA-BLS entirely in the real domain while preserving its exact decision function, achieving up to 75% fewer multiplications and over 60% fewer additions. A rigorous theoretical analysis proves the mathematical equivalence between CA-BLS and ECA-BLS, ensuring zero theoretical loss. Extensive experiments on 26 benchmark datasets from the UCI and KEEL repositories demonstrate that ECA-BLS consistently outperforms classical BLS and recent state-of-the-art randomized neural networks in accuracy, average rank, and statistical significance, establishing augmented second-order modeling as a critical and previously missing dimension of BLS research.
A. Rahaman, A. Quadir, M. Sajid +2
Department of Mathematics Indian Institute of Technology Indore Indore, India
The Broad Learning System (BLS) has been widely used for data classification and is based on a layer-by-layer feed-forward structure. However, it gives the same importance to all data points, which reduces its effectiveness on real-world datasets with noise and outliers. In addition, it does not consider the geometric structure of the data and has limitations in handling data from multiple sources. To address these challenges, we propose a Multi-View Graph-Embedded Intuitionistic Fuzzy Broad Learning System (MVGIFBLS) that integrates multi-view learning, graph embedding, and intuitionistic fuzzy theory into the BLS framework. This design enables the model to combine information from multiple sources and learn more discriminative representations. Graph embedding captures the geometric relationships among samples and improves class separation through intrinsic and penalty subspaces based on local Fisher discriminant analysis. Intuitionistic fuzzy theory enhances robustness to noise, while kernel-based neighborhood analysis captures local data structures. We evaluate the proposed framework on several UCI, KEEL, and AwA benchmark datasets using comparative evaluation, Gaussian feature noise analysis, ablation studies, and statistical analysis. The results demonstrate that each component contributes positively to the overall framework and that the proposed MVGIFBLS consistently achieves higher Area Under the Curve (AUC) scores and maintains robust performance under Gaussian feature noise.
Yogesh Kumar, Manju, Mudasir Ganaie
Department of Computer Science and Engineering, Indian Institute of Technology Ropar, Rupnagar, 140001, Punjab, India.