Generalization Error Bounds for Iterative Recovery Algorithms Unfolded as Neural Networks
This work provides theoretical guarantees for iterative recovery algorithms unfolded as neural networks, addressing generalization in sparse reconstruction, but it is incremental as it builds on existing methods like LISTA.
The paper introduces a general class of neural networks for sparse reconstruction from linear measurements, deriving generalization bounds via Rademacher complexity that yield sample complexity estimates depending linearly on parameters and depth.
Motivated by the learned iterative soft thresholding algorithm (LISTA), we introduce a general class of neural networks suitable for sparse reconstruction from few linear measurements. By allowing a wide range of degrees of weight-sharing between the layers, we enable a unified analysis for very different neural network types, ranging from recurrent ones to networks more similar to standard feedforward neural networks. Based on training samples, via empirical risk minimization we aim at learning the optimal network parameters and thereby the optimal network that reconstructs signals from their low-dimensional linear measurements. We derive generalization bounds by analyzing the Rademacher complexity of hypothesis classes consisting of such deep networks, that also take into account the thresholding parameters. We obtain estimates of the sample complexity that essentially depend only linearly on the number of parameters and on the depth. We apply our main result to obtain specific generalization bounds for several practical examples, including different algorithms for (implicit) dictionary learning, and convolutional neural networks.