An $$L_{2,1}$$-regularized cascaded broad learning framework for robust anomaly detection in complex industrial metrology

To address the complexity, high dimensionality, and severe data imbalance in industrial metrology, we propose an $$L_{2,1}$$-Regularized Cascaded Broad Learning Framework (L21RCBLS) for robust anomaly detection. Our study makes three key contributions: (i) a cascaded feature structure with residual connections is designed to enhance sequential representation while mitigating gradient instability and parameter forgetting; (ii) a sparsity-inducing $$L_{2,1}$$-norm optimization strategy is introduced to improve robustness against ubiquitous industrial noise and scattered outliers; and (iii) the mathematical convergence of the proposed iterative optimization algorithm is theoretically established. Extensive experiments on real-world industrial datasets (over 16,600 samples) demonstrate that L21RCBLS consistently outperforms eight baseline models, including CNN, BiGRU, and pNBEBLS. For three critical product dimensions (Size 1 to 3), the model achieves optimal performance with Mean Squared Error (MSE) as low as (0.000152, 0.000152, 0.000098) and $$R^2$$ scores exceeding (0.914520, 0.943989, 0.921815). Ablation studies further confirm that the synergy of residual cascading and $$L_{2,1}$$-regularization is vital for maintaining high precision under limited and skewed data conditions, offering a computationally efficient solution for real-time quality control.