Theory of periodic convolutional neural network
This work expands the mathematical foundation of CNN approximation theory and highlights architectures relevant for domains like image analysis on wrapped domains, physics-informed learning, and materials science, though it is primarily theoretical and incremental in nature.
The paper tackles the problem of characterizing the expressive power of convolutional neural networks by introducing a periodic CNN architecture with periodic boundary conditions, and proves a rigorous approximation theorem showing it can approximate ridge functions with d-1 linear variables in d-dimensional spaces, which is impossible for lower-dimensional settings.
We introduce a novel convolutional neural network architecture, termed the \emph{periodic CNN}, which incorporates periodic boundary conditions into the convolutional layers. Our main theoretical contribution is a rigorous approximation theorem: periodic CNNs can approximate ridge functions depending on $d-1$ linear variables in a $d$-dimensional input space, while such approximation is impossible in lower-dimensional ridge settings ($d-2$ or fewer variables). This result establishes a sharp characterization of the expressive power of periodic CNNs. Beyond the theory, our findings suggest that periodic CNNs are particularly well-suited for problems where data naturally admits a ridge-like structure of high intrinsic dimension, such as image analysis on wrapped domains, physics-informed learning, and materials science. The work thus both expands the mathematical foundation of CNN approximation theory and highlights a class of architectures with surprising and practically relevant approximation capabilities.