{"ID":2867000,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2509.18744","arxiv_id":"2509.18744","title":"Theory of periodic convolutional neural network","abstract":"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.","short_abstract":"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 va...","url_abs":"https://arxiv.org/abs/2509.18744","url_pdf":"https://arxiv.org/pdf/2509.18744v2","authors":"[\"Yuqing Liu\"]","published":"2025-09-23T07:43:02Z","proceeding":"cs.LG","tasks":"[\"cs.LG\"]","methods":"[\"Convolutional Neural Network\"]","has_code":false}