用凸优化方法解决神经网络正则化图像重建难题
A primal-dual algorithm for image reconstruction with input-convex neural network regularizers
- 将非光滑神经网络正则化重构问题转化为可解的凸优化问题
- 在多个成像任务中优于传统梯度与加速方法
- 支持正则化器自身训练，适合需要端到端学习的场景
我们研究数据驱动变分重建框架中的优化问题，其中正则项由输入凸神经网络（ICNN）参数化。尽管基于梯度的方法常用于此类问题，但难以有效处理非光滑问题，导致收敛缓慢；且神经网络的嵌套结构使标准非光滑优化技术（如近端算法）难以应用。为此，我们重新构建问题，消除网络嵌套结构，通过将该重构与激活函数的上图投影关联，将原问题转化为可高效求解的凸优化问题，并证明其与原始变分问题等价。在多个成像任务上的实验表明，该方法不仅在平滑设置下超越子梯度法甚至加速方法，还支持正则化器本身的训练。
原文摘要 · Abstract (English)
We address the optimization problem in a data-driven variational reconstruction framework, where the regularizer is parameterized by an input-convex neural network (ICNN). While gradient-based methods are commonly used to solve such problems, they struggle to effectively handle non-smooth problems which often leads to slow convergence. Moreover, the nested structure of the neural network complicates the application of standard non-smooth optimization techniques, such as proximal algorithms. To overcome these challenges, we reformulate the problem and eliminate the network's nested structure. By relating this reformulation to epigraphical projections of the activation functions, we transform the problem into a convex optimization problem that can be efficiently solved using a primal-dual algorithm. We also prove that this reformulation is equivalent to the original variational problem. Through experiments on several imaging tasks, we show that the proposed approach not only outperforms subgradient methods and even accelerated methods in the smooth setting, but also facilitates the training of the regularizer itself.
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