Algorithm unfolding or unrolling is the technique of constructing a deep neural network (DNN) from an iterative algorithm. Unrolled DNNs often provide better interpretability and superior empirical performance over standard DNNs in signal estimation tasks. An important theoretical question, which has only recently received attention, is the development of generalization error bounds for unrolled DNNs. These bounds deliver theoretical and practical insights into the performance of a DNN on empirical datasets that are distinct from, but sampled from, the probability density generating the DNN training data. In this paper, we develop novel generalization error bounds for a class of unrolled DNNs that are informed by a compound Gaussian prior. These compound Gaussian networks have been shown to outperform comparative standard and unfolded deep neural networks in compressive sensing and tomographic imaging problems. The generalization error bound is formulated by bounding the Rademacher complexity of the class of compound Gaussian network estimates with Dudley's integral. Under realistic conditions, we show that, at worst, the generalization error scales $\mathcal{O}(n\sqrt{\ln(n)})$ in the signal dimension and $\mathcal{O}(($Network Size$)^{3/2})$ in network size.

翻译：算法展开或解折叠是一种从迭代算法构建深度神经网络的技术。在信号估计任务中，展开式深度神经网络通常比标准深度神经网络具有更好的可解释性和更优的经验性能。一个重要但近期才受到关注的理论问题是展开式深度神经网络泛化误差界的开发。这些边界为深度神经网络在独立于训练数据但采样自相同概率密度的经验数据集上的性能提供了理论与实践的洞察。本文针对一类受复合高斯先验启发的展开式深度神经网络，提出了新的泛化误差界。实验表明，这类复合高斯网络在压缩感知与断层成像问题中优于对比的标准及展开式深度神经网络。该泛化误差界通过Dudley积分对复合高斯网络估计类别的Rademacher复杂度进行约束而构建。在现实条件下，我们证明泛化误差在最坏情况下随信号维度以$\mathcal{O}(n\sqrt{\ln(n)})$的阶数缩放，随网络规模以$\mathcal{O}(($网络规模$)^{3/2})$的阶数缩放。