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Compositional Function Spaces for Deep Learning
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AI
DOI：10.1137/25M1802948.png)
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Abstract
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We present a variational framework for studying functions learned by deep neural networks with rectified linear unit nonlinearities. We introduce a function space built from compositions of functions of second-order Radon-domain bounded variation. The compositional form of these functions captures the structure of deep neural networks. We prove a representer theorem that shows that deep neural networks with finite width solve regularized data-fitting problems over this space. The critical width is controlled by the square of the number of training data. This perspective explains the effect of weight-decay regularization in neural network training, the importance of skip connections, and the role of sparsity in neural networks. By considering the function-space perspective, we provide sharp links between deep learning and variational methods.
Keywords:
deep learning
neural networks
regularization
representer theorem
sparsity
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6.1
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888
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