The recently proposed fully-connected tensor network (FCTN) decomposition has demonstrated significant advantages in correlation characterization and transpositional invariance, and has achieved notable achievements in multi-dimensional data processing and analysis. However, existing multi-dimensional data recovery methods leveraging FCTN decomposition still have room for further enhancement, particularly in computational efficiency and modeling capability. To address these issues, we first propose a FCTN-based generalized nonconvex regularization paradigm from the perspective of gradient mapping. Then, reliable and scalable multi-dimensional data recovery models are investigated, where the model formulation is shifted from unquantized observations to coarse-grained quantized observations. Based on the alternating direction method of multipliers (ADMM) framework, we derive efficient optimization algorithms with convergence guarantees to solve the formulated models. To alleviate the computational bottleneck encountered when processing large-scale multi-dimensional data, fast and efficient randomized compression algorithms are devised in virtue of sketching techniques in numerical linear algebra. These dimensionality-reduction techniques serve as the computational acceleration core of our proposed algorithm framework. Theoretical results on approximation error upper bounds and convergence analysis for the proposed method are derived. Extensive numerical experiments illustrate the effectiveness and superiority of the proposed algorithm over other state-of-the-art methods in terms of quantitative metrics, visual quality, and running time.

翻译：最近提出的全连接张量网络（FCTN）分解在相关性表征与转置不变性方面展现出显著优势，并在多维数据处理与分析中取得了突出成果。然而，现有基于FCTN分解的多维数据恢复方法在计算效率与建模能力方面仍有进一步提升空间。针对这些问题，我们首先从梯度映射角度提出了一种基于FCTN的广义非凸正则化范式。随后，研究了可靠且可扩展的多维数据恢复模型，将模型构建从非量化观测转向粗粒度量化观测。基于交替方向乘子法（ADMM）框架，我们推导出具有收敛保证的高效优化算法来求解所构建的模型。为缓解处理大规模多维数据时遇到的计算瓶颈，借助数值线性代数中的草图技术设计了快速高效的随机压缩算法。这些降维技术构成我们提出算法框架的计算加速核心。我们推导了所提方法的近似误差上界理论结果与收敛性分析。大量数值实验表明，所提算法在量化指标、视觉质量与运行时间方面均优于其他先进方法。