Deep Learning Meets Sparse Regularization: A Signal Processing Perspective

Abstract

Deep learning has been wildly successful in practice and most state-of-the-art machine learning methods are based on neural networks. Lacking, however, is a rigorous mathematical theory that adequately explains the amazing performance of deep neural networks. In this article, we present a relatively new mathematical framework that provides the beginning of a deeper understanding of deep learning. This framework precisely characterizes the functional properties of neural networks that are trained to fit to data. The key mathematical tools which support this framework include transform-domain sparse regularization, the Radon transform of computed tomography, and approximation theory, which are all techniques deeply rooted in signal processing. This framework explains the effect of weight decay regularization in neural network training, the use of skip connections and low-rank weight matrices in network architectures, the role of sparsity in neural networks, and explains why neural networks can perform well in high-dimensional problems.

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Cited by in corpus (5)

• Distributional Extension and Invertibility of the $k$-Plane Transform and Its Dual
• Weighted variation spaces and approximation by shallow ReLU networks
• Smoothing the Edges: Smooth Optimization for Sparse Regularization using Hadamard Overparametrization
• Function-Space Optimality of Neural Architectures with Multivariate Nonlinearities
• An Overview of Low-Rank Structures in the Training and Adaptation of Large Models