We establish a layer-wise parameterization for 1D convolutional neural networks (CNNs) with built-in end-to-end robustness guarantees. In doing so, we use the Lipschitz constant of the input-output mapping characterized by a CNN as a robustness measure. We base our parameterization on the Cayley transform that parameterizes orthogonal matrices and the controllability Gramian of the state space representation of the convolutional layers. The proposed parameterization by design fulfills linear matrix inequalities that are sufficient for Lipschitz continuity of the CNN, which further enables unconstrained training of Lipschitz-bounded 1D CNNs. Finally, we train Lipschitz-bounded 1D CNNs for the classification of heart arrythmia data and show their improved robustness.

翻译：我们为具有内在端到端鲁棒性保障的一维卷积神经网络（CNN）建立了一种逐层参数化方法。具体而言，我们将CNN输入-输出映射的Lipschitz常数作为鲁棒性度量，并基于正交矩阵参数化的Cayley变换以及卷积层状态空间表示的可控性格拉姆矩阵构建参数化方案。该参数化方法通过设计满足保证CNN Lipschitz连续性的线性矩阵不等式，进而实现了对Lipschitz有界一维CNN的无约束训练。最后，我们训练了用于心律不齐数据分类的Lipschitz有界一维CNN，并验证了其鲁棒性的提升。