Fourier Multi-Component and Multi-Layer Neural Networks: Unlocking High-Frequency Potential

The two most critical ingredients of a neural network are its structure and the activation function employed, and more importantly, the proper alignment of these two that is conducive to the effective representation and learning in practice. In this work, we introduce a surprisingly effective synergy, termed the Fourier Multi-Component and Multi-Layer Neural Network (FMMNN), and demonstrate its surprising adaptability and efficiency in capturing high-frequency components. First, we theoretically establish that FMMNNs have exponential expressive power in terms of approximation capacity. Next, we analyze the optimization landscape of FMMNNs and show that it is significantly more favorable compared to fully connected neural networks. Finally, systematic and extensive numerical experiments validate our findings, demonstrating that FMMNNs consistently achieve superior accuracy and efficiency across various tasks, particularly impressive when high-frequency components are present.

Structured and Balanced Multi-component and Multi-layer Neural Networks

In this work, we propose a balanced multi-component and multi-layer neural network (MMNN) structure to approximate functions with complex features with both accuracy and efficiency in terms of degrees of freedom and computation cost. The main idea is motivated by a multi-component, each of which can be approximated effectively by a single-layer network, and multi-layer decomposition in a "divide-and-conquer" type of strategy to deal with a complex function. While an easy modification to fully connected neural networks (FCNNs) or multi-layer perceptrons (MLPs) through the introduction of balanced multi-component structures in the network, MMNNs achieve a significant reduction of training parameters, a much more efficient training process, and a much improved accuracy compared to FCNNs or MLPs. Extensive numerical experiments are presented to illustrate the effectiveness of MMNNs in approximating high oscillatory functions and its automatic adaptivity in capturing localized features.

Robustness of the data-driven approach in limited angle tomography

The limited angle Radon transform is notoriously difficult to invert due to the ill-posedness. In this work, we give a mathematical explanation that the data-driven approach based on deep neural networks can reconstruct more information in a stable way compared to traditional methods.

Why Shallow Networks Struggle with Approximating and Learning High Frequency: A Numerical Study

In this work, a comprehensive numerical study involving analysis and experiments shows why a two-layer neural network has difficulties handling high frequencies in approximation and learning when machine precision and computation cost are important factors in real practice. In particular, the following fundamental computational issues are investigated: (1) the best accuracy one can achieve given a finite machine precision, (2) the computation cost to achieve a given accuracy, and (3) stability with respect to perturbations. The key to the study is the spectral analysis of the corresponding Gram matrix of the activation functions which also shows how the properties of the activation function play a role in the picture.