Early Stopping of Untrained Convolutional Neural Networks
CoRR(2024)
摘要
In recent years, new regularization methods based on (deep) neural networks
have shown very promising empirical performance for the numerical solution of
ill-posed problems, such as in medical imaging and imaging science. Due to the
nonlinearity of neural networks, these methods often lack satisfactory
theoretical justification. In this work, we rigorously discuss the convergence
of a successful unsupervised approach that utilizes untrained convolutional
neural networks to represent solutions to linear ill-posed problems. Untrained
neural networks have particular appeal for many applications because they do
not require paired training data. The regularization property of the approach
relies solely on the architecture of the neural network instead. Due to the
vast over-parameterization of the employed neural network, suitable early
stopping is essential for the success of the method. We establish that the
classical discrepancy principle is an adequate method for early stopping of
two-layer untrained convolutional neural networks learned by gradient descent,
and furthermore, it yields an approximation with minimax optimal convergence
rates. Numerical results are also presented to illustrate the theoretical
findings.
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