论文标题

均匀凸出神经网络和非平稳迭代网络Tikhonov(INETT)方法

Uniformly convex neural networks and non-stationary iterated network Tikhonov (iNETT) method

论文作者

Bianchi, Davide, Lai, Guanghao, Li, Wenbin

论文摘要

我们提出了一种非平稳的迭代网络Tikhonov(INETT)方法,用于解决方案不良问题。 INETT采用深层神经网络来构建数据驱动的正常化程序,并避免了估计最佳正则化参数的艰巨任务。为了实现INETT的理论融合,我们引入了统一的凸神经网络以构建数据驱动的正常器。提出了严格的理论和详细的算法,用于构建凸面和均匀的凸神经网络。特别是,在一般的神经网络体系结构的情况下,我们规定了足够的条件,以实现训练有素的神经网络,该神经网络是组成部分的凸面或均匀凸的。此外,我们提供了实现现代U-NET体系结构中凸度和均匀凸度的具体例子。借助凸的工具和均匀的凸神经网络,开发了INETT算法,并提供了严格的合并分析。最后,我们在2D计算机断层扫描中显示了INETT算法的应用,其中数值示例说明了所提出算法的功效。

We propose a non-stationary iterated network Tikhonov (iNETT) method for the solution of ill-posed inverse problems. The iNETT employs deep neural networks to build a data-driven regularizer, and it avoids the difficult task of estimating the optimal regularization parameter. To achieve the theoretical convergence of iNETT, we introduce uniformly convex neural networks to build the data-driven regularizer. Rigorous theories and detailed algorithms are proposed for the construction of convex and uniformly convex neural networks. In particular, given a general neural network architecture, we prescribe sufficient conditions to achieve a trained neural network which is component-wise convex or uniformly convex; moreover, we provide concrete examples of realizing convexity and uniform convexity in the modern U-net architecture. With the tools of convex and uniformly convex neural networks, the iNETT algorithm is developed and a rigorous convergence analysis is provided. Lastly, we show applications of the iNETT algorithm in 2D computerized tomography, where numerical examples illustrate the efficacy of the proposed algorithm.

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