Deformation Stability of Deep Convolutional Neural Networks on Sobolev Spaces
Deformation Stability of Deep Convolutional Neural Networks on Sobolev Spaces
复制标题
Sobolev 空间上深度卷积神经网络的变形稳定性
DOI：
10.1109/icassp.2018.8462158
复制
发表时间：
2018
期刊：
影响因子：
--
通讯作者：
H. Boche
中科院分区：
文献类型：
--
作者：
M. Koller;Johannes Grobmann;U. Mönich;H. Boche
我们的工作是基于最近引入的深度卷积神经网络（DCNNs）的数学理论。结果表明，对于带限输入函数的变形，DCNN是稳定的。在本文中，我们推广了这一结果：证明了Sobolev空间上的变形稳定性。进一步，我们展示了整个输入空间L2（Rd）的变形稳定性的弱形式。DCNNs的基本组成部分是半离散帧。对于实际应用，有必要进行具体的选择。因此，我们提出了一种基于有界统一分割（bupu）的半离散框架的构造方法，并给出了一个使用b样条的具体例子。
Our work is based on a recently introduced mathematical theory of deep convolutional neural networks (DCNNs). It was shown that DCNN s are stable with respect to deformations of bandlimited input functions. In the present paper, we generalize this result: We prove deformation stability on Sobolev spaces. Further, we show a weak form of deformation stability for the whole input space L2(Rd). The basic components of DCNNs are semi-discrete frames. For practical applications, a concrete choice is necessary. Therefore, we conclude our work by suggesting a construction method for semi-discrete frames based on bounded uniform partitions of unity (BUPUs) and give a specific example that uses B-splines.