面向多维数据恢复的重参数化张量环函数分解
Reparameterized Tensor Ring Functional Decomposition for Multi-Dimensional Data Recovery
把张量环分解的因子用隐式神经表示参数化，使分解能作用于非网格数据（如点云），并通过频率域分析指出因子谱结构限制高频建模，再用「可学习潜张量×固定基」的重参数化改善训练动力学——这套思路可直接迁移到任意基于 INR 的连续信号重建与逆问题求解中。
原文依据 the spectral structure of TR factors determines the frequency composition of the reconstructed tensor and limits the high-frequency modeling capacity
摘要译文
由模型翻译 · 术语统一 · 请对照原文核对关键结论张量环（TR）分解是高阶数据建模的有力工具，但其本质上局限于定义在固定网格上的离散形式。本文提出一种同时适用于网格数据与非网格数据的 TR 函数分解，其中各因子由隐式神经表示（INR）参数化。然而，优化这一连续框架以捕捉细粒度细节本身十分困难。通过频率域分析，我们证明 TR 因子的谱结构决定了重建张量的频率组成，并限制了高频建模能力。为缓解这一问题，我们提出重参数化的 TR 函数分解，其中每个 TR 因子是可学习潜张量与固定基的结构化组合。理论上证明，该重参数化能够改善 TR 因子学习的训练动力学。我们进一步为固定基推导出有原则的初始化方案，并证明了所提模型的 Lipschitz 连续性。在图像修复、去噪、超分辨率和点云恢复上的大量实验表明，本方法相较现有方法取得了一致更优的性能。代码见 https://github.com/YangyangXu2002/RepTRFD。
英文原文摘要 1305 字符
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.