Deep neural networks are often used to implement powerful generative models for real-world data. Notable applications include image denoising, as well as other classical inverse problems like compressed sensing and super-resolution. To provide a rigorous but simplified analysis of generative models, in this work, we introduce an elegant theoretical framework based on spherical harmonics, namely \textbf{SUNLayer}. Our theoretical framework identifies explicit conditions on activation functions that guarantee denoising under local optimization. Numerical experiments examine the theoretical properties on commonly used activation functions and demonstrate their stable denoising performance.

翻译：深度神经网络常用于实现针对现实世界数据的强大生成模型。显著应用包括图像去噪，以及压缩感知和超分辨率等其他经典逆问题。为对生成模型提供严谨而简化的分析，本研究引入了一个基于球谐函数的优雅理论框架，即\textbf{SUNLayer}。该理论框架明确了激活函数需满足的显式条件，以保证局部优化下的去噪效果。数值实验检验了常用激活函数的理论特性，并验证了其稳定的去噪性能。