🤖 AI Summary
This work proposes a continuous tensor ring (TR) function decomposition method based on implicit neural representations (INRs), addressing the limitations of traditional TR decomposition, which is confined to fixed-grid data and struggles with non-grid high-dimensional signals and high-frequency detail recovery. By integrating INRs into the TR framework for the first time, the approach enables modeling at arbitrary sampling points. A reparameterization strategy expresses each TR factor as a structured combination of learnable latent tensors and fixed basis functions, enhancing both high-frequency modeling capacity and training stability. Theoretical analysis demonstrates improved optimization dynamics and Lipschitz continuity. Experiments show that the method significantly outperforms existing approaches in tasks such as image inpainting, denoising, super-resolution, and point cloud reconstruction, exhibiting superior generalization and multidimensional signal reconstruction capabilities.
📝 Abstract
Tensor Ring (TR) decomposition is a powerful tool for high-order data modeling, but is inherently restricted to discrete forms defined on fixed meshgrids. In this work, we propose a TR functional decomposition for both meshgrid and non-meshgrid data, where factors are parameterized by Implicit Neural Representations (INRs). However, optimizing this continuous framework to capture fine-scale details is intrinsically difficult. Through a frequency-domain analysis, we demonstrate that the spectral structure of TR factors determines the frequency composition of the reconstructed tensor and limits the high-frequency modeling capacity. To mitigate this, we propose a reparameterized TR functional decomposition, in which each TR factor is a structured combination of a learnable latent tensor and a fixed basis. This reparameterization is theoretically shown to improve the training dynamics of TR factor learning. We further derive a principled initialization scheme for the fixed basis and prove the Lipschitz continuity of our proposed model. Extensive experiments on image inpainting, denoising, super-resolution, and point cloud recovery demonstrate that our method achieves consistently superior performance over existing approaches. Code is available at https://github.com/YangyangXu2002/RepTRFD.