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 this https URL.
Abstract (translated)
张量环（TR）分解是高阶数据分析的强大工具，但本质上局限于在固定网格上的离散形式。在这项工作中，我们提出了一种适用于网格数据和非网格数据的TR函数分解方法，其中因子由隐式神经表示(INR)参数化。然而，优化这种连续框架以捕捉细粒度细节在本质上是困难的。通过频域分析，我们展示了TR因子的谱结构决定了重构张量的频率构成，并限制了高频建模的能力。为了解决这个问题，我们提出了一个重新参数化的TR函数分解方法，在这种方法中每个TR因子是由可学习的潜在张量和固定基底有结构地组合而成。理论上证明，这种重参数化可以改善TR因子的学习动态过程。此外，我们还推导出了一种用于固定基底的原则性初始化方案，并证明了所提出模型的Lipschitz连续性。在图像修复、去噪、超分辨率和点云恢复等广泛的实验中表明，我们的方法比现有方法达到了一致且优越的表现水平。代码可在提供的URL处获取。
URL
https://arxiv.org/abs/2603.01034
