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Deep neural networks (DNNs) have showcased their remarkable precision in approximating smooth functions. However, they suffer from the spectral bias, wherein DNNs typically exhibit a tendency to prioritize the learning of lower-frequency components of a function, struggling to effectively capture its high-frequency features. This paper is to address this issue. Notice that a function having only low frequency components may be well-represented by a shallow neural network (SNN), a network having only a few layers. By observing that composition of low frequency functions can effectively approximate a high-frequency function, we propose to learn a function containing high-frequency components by composing several SNNs, each of which learns certain low-frequency information from the given data. We implement the proposed idea by exploiting the multi-grade deep learning (MGDL) model, a recently introduced model that trains a DNN incrementally, grade by grade, a current grade learning from the residue of the previous grade only an SNN (with trainable parameters) composed with the SNNs (with fixed parameters) trained in the preceding grades as features. We apply MGDL to synthetic, manifold, colored images, and MNIST datasets, all characterized by presence of high-frequency features. Our study reveals that MGDL excels at representing functions containing high-frequency information. Specifically, the neural networks learned in each grade adeptly capture some low-frequency information, allowing their compositions with SNNs learned in the previous grades effectively representing the high-frequency features. Our experimental results underscore the efficacy of MGDL in addressing the spectral bias inherent in DNNs. By leveraging MGDL, we offer insights into overcoming spectral bias limitation of DNNs, thereby enhancing the performance and applicability of deep learning models in tasks requiring the representation of high-frequency information. This study confirms that the proposed method offers a promising solution to address the spectral bias of DNNs. The code is available on GitHub: Addressing Spectral Bias via MGDL. 

Abstract We studyinexactfixed-point proximity algorithms for solving a class of sparse regularization problems involving the$$\ell _0$$ ${\ell }_0$norm. Specifically, the$$\ell _0$$ ${\ell }_0$model has an objective function that is the sum of a convex fidelity term and a Moreau envelope of the$$\ell _0$$ ${\ell }_0$norm regularization term. Such an$$\ell _0$$ ${\ell }_0$model is non-convex. Existing exact algorithms for solving the problems require the availability of closed-form formulas for the proximity operator of convex functions involved in the objective function. When such formulas are not available, numerical computation of the proximity operator becomes inevitable. This leads to inexact iteration algorithms. We investigate in this paper how the numerical error for every step of the iteration should be controlled to ensure global convergence of the inexact algorithms. We establish a theoretical result that guarantees the sequence generated by the proposed inexact algorithm converges to a local minimizer of the optimization problem. We implement the proposed algorithms for three applications of practical importance in machine learning and image science, which include regression, classification, and image deblurring. The numerical results demonstrate the convergence of the proposed algorithm and confirm that local minimizers of the$$\ell _0$$ ${\ell }_0$models found by the proposed inexact algorithm outperform global minimizers of the corresponding$$\ell _1$$ ${\ell }_1$models, in terms of approximation accuracy and sparsity of the solutions.