DALR: A Density-Adaptive Laplacian Regularization Method for Enhancing Model Robustness and Generalization

The sensitivity of neural networks to input perturbations significantly undermines their generalization performance and robustness in complex real-world environments. To address this issue, This paper proposes an adaptive densityweighted Laplacian regularization method, termed DALR (Density-Adaptive Lanczos Laplacian Regularization). This method aims to enhance model stability and generalization under perturbation by efficiently estimating the second-order sensitivity of the model to input variations and adaptively applying local density-weighted regularization. DALR integrates the efficient spectral trace estimation of the Hessian matrix of the loss function via the Lanczos algorithm and introduces an adaptive weighting mechanism based on data-manifold density perception. Unlike conventional Laplacian regularization which typically relies on graph-based affinity or similarity matrices, DALR implicitly captures local smoothness through the spectral properties of the loss Hessian, providing a data-driven and model-aware approximation of Laplacian behavior. Consequently, the model enforces stronger constraints in low-density, high-risk regions of the feature space, thereby mitigating overfitting in these areas. Experiments conducted on CIFAR-10-C at severity levels 3 and 5 demonstrated that, when combined with L2 regularization, DALR achieved accuracies of 79.21% and 60.01%, respectively. These results indicate that DALR not only effectively improves model accuracy and robustness under severe perturbations but also offers new insights into model optimization for future complex visual tasks and cross-domain applications.