Abstract
Multi-dimensional data completion is a critical problem in computational sciences, particularly in domains such as computer vision, signal processing, and scientific computing. Existing methods typically leverage either global low-rank approximations or local smoothness regularization, but each suffers from notable limitations: low-rank methods are computationally expensive and may disrupt intrinsic data structures, while smoothness-based approaches often require extensive manual parameter tuning and exhibit poor generalization. In this paper, we propose a novel Low-Rank Tucker Representation (LRTuckerRep) model that unifies global and local prior modeling within a Tucker decomposition. Specifically, LRTuckerRep encodes low rankness through a self-adaptive weighted nuclear norm on the factor matrices and a sparse Tucker core, while capturing smoothness via a parameter-free Laplacian-based regularization on the factor spaces. To efficiently solve the resulting nonconvex optimization problem, we develop two iterative algorithms with provable convergence guarantees. Extensive experiments on multi-dimensional image inpainting and traffic data imputation demonstrate that LRTuckerRep achieves superior completion accuracy and robustness under high missing rates compared to baselines.
Abstract (translated)
多维数据补全在计算科学中是一个关键问题，尤其是在计算机视觉、信号处理和科学计算等领域。现有方法通常利用全局低秩近似或局部光滑性正则化，但每种方法都存在显著的局限性：低秩方法计算成本高且可能破坏内在的数据结构，而基于光滑性的方法往往需要大量的手动参数调整，并表现出较差的泛化能力。在本文中，我们提出了一种新颖的低秩Tucker表示（LRTuckerRep）模型，该模型在一个Tucker分解框架内统一了全局和局部先验建模。具体来说，LRTuckerRep通过因子矩阵上的自适应加权核范数以及稀疏的Tucker核心来编码低秩性，同时使用基于拉普拉斯算子的无参数正则化来捕捉光滑性。为了有效地解决由此产生的非凸优化问题，我们开发了两个迭代算法，并证明了它们具有收敛保证。在多维图像修复和交通数据插补上的广泛实验表明，在高缺失率情况下，LRTuckerRep比基准方法表现出更优越的完成准确性和鲁棒性。
URL
https://arxiv.org/abs/2508.03755
