发表机构
Southwest University（西南大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对低秩张量补全中无加权谱正则化过度衰减主导分量的问题，提出两种加权Schatten-p因子化模型WSpTFI和WSpTFII，分别采用ADMM和IRLS-BSUM求解，在多个任务上验证了鲁棒性和重建质量。
AI 中文摘要
低秩张量因子化为从不完整和损坏的观测中补全多维数据提供了一个灵活的框架。然而，无加权的谱正则化器对奇异分量施加共同的收缩轮廓，这可能过度衰减主导的低秩分量，而因子化变体要么缺乏分量特定的加权，要么需要昂贵的奇异值分解（SVD）。本文在张量-张量积（t-product）框架下提出了两种加权Schatten-$p$张量因子化模型，分别称为\WSpTFI{}和\WSpTFII{}，以解决这些局限性。\WSpTFI{}由因子化加权张量Schatten-$p$范数恒等式驱动，并允许灵活的、可能不对称的因子指数。\WSpTFII{}从变换域列对能量构造正则化器，实现无SVD的主因子更新和用于减少冗余秩分量的列剪枝机制。本文进一步为\WSpTFI{}开发了一种迭代重加权交替方向乘子法（ADMM）型方案，为\WSpTFII{}开发了一种迭代重加权最小二乘（IRLS）-块连续上界最小化（BSUM）方案。理论分析建立了加权因子化关系，并为\WSpTFI{}提供了条件极限Karush-Kuhn-Tucker（KKT）表征。对于\WSpTFII{}，实际的阻尼二次块更新为固定$\delta$平滑因子目标提供了定量的充分下降机制。这意味着渐近正则性，并且固定维尾部的每个累积点都是平稳的。在合成张量补全、彩色图像恢复、高光谱图像修复和印刷电路板缺陷检测上的实验证明了在各种退化条件下的竞争性重建质量和鲁棒性。
英文摘要
Low-rank tensor factorization provides a flexible framework for completing multidimensional data from incomplete and corrupted observations. However, unweighted spectral regularizers impose a common shrinkage profile across singular components, which may excessively attenuate dominant low-rank components, and factorized variants either lack component-specific weighting or require costly singular value decompositions (SVDs). This paper proposes two weighted Schatten-$p$ tensor factorization models, termed \WSpTFI{} and \WSpTFII{}, under the tensor-tensor product (t-product) framework to address these limitations. \WSpTFI{} is motivated by a factorized weighted tensor Schatten-$p$ norm identity and permits flexible, possibly asymmetric factor exponents. \WSpTFII{} constructs a regularizer from transform-domain column-pair energies, yielding SVD-free main factor updates and a column-pruning mechanism for reducing redundant rank components. This paper further develops an iteratively reweighted alternating direction method of multipliers (ADMM)-type scheme for \WSpTFI{} and an iteratively reweighted least squares (IRLS)--block successive upper-bound minimization (BSUM) scheme for \WSpTFII{}. Theoretical analysis establishes the weighted factorization relation and provides a conditional limiting Karush--Kuhn--Tucker (KKT) characterization for \WSpTFI{}. For \WSpTFII{}, the actual damped quadratic block updates yield a quantitative sufficient-decrease mechanism for the fixed-$δ$ smoothed factor objective. This implies asymptotic regularity, and every accumulation point of the fixed-dimensional tail is stationary. Experiments on synthetic tensor completion, color-image restoration, hyperspectral inpainting, and printed-circuit-board defect detection demonstrate competitive reconstruction quality and robustness under various degradation conditions.
