面向具备可保证鲁棒性的张量PCA的可学习缩放梯度下降方法
Learnable Scaled Gradient Descent for Guaranteed Robust Tensor PCA
AI总结：
针对现有鲁棒张量PCA方法依赖高计算成本张量核范数的问题，提出基于t-SVD框架的RTPCA-SGD方法，证明其线性收敛保证，并提出可学习自监督深度展开模型优化参数，实现性能与效率的双重提升。
摘要：
鲁棒张量主成分分析（RTPCA）旨在从多维数据中分离低秩分量与稀疏分量，是信号处理和计算机视觉领域的核心技术。近年来新兴的张量奇异值分解（t-SVD）因相比传统矩阵SVD能更好地捕捉张量的低秩结构而受到广泛关注。然而现有方法通常依赖计算成本高昂的张量核范数（TNN），限制了其在真实张量场景下的可扩展性。为解决该问题，我们首次在t-SVD框架下探索高效的缩放梯度下降（SGD）方法，提出RTPCA-SGD算法。理论上，我们在温和假设下严格证明了RTPCA-SGD的恢复保证：参数选择得当时，该方法能以恒定速率线性收敛到真实低秩张量，且收敛速率与条件数无关。为提升实际适用性，我们进一步提出一种可学习的自监督深度展开模型，可实现有效的参数学习。在合成数据集与真实数据集上的数值实验表明，所提方法性能优异且保持了具竞争力的计算效率，尤其耗时低于RTPCA-TNN。
英文摘要：
Robust tensor principal component analysis (RTPCA) aims to separate the low-rank and sparse components from multi-dimensional data, making it an essential technique in the signal processing and computer vision fields. Recently emerging tensor singular value decomposition (t-SVD) has gained considerable attention for its ability to better capture the low-rank structure of tensors compared to traditional matrix SVD. However, existing methods often rely on the computationally expensive tensor nuclear norm (TNN), which limits their scalability for real-world tensors. To address this issue, we explore an efficient scaled gradient descent (SGD) approach within the t-SVD framework for the first time, and propose the RTPCA-SGD method. Theoretically, we rigorously establish the recovery guarantees of RTPCA-SGD under mild assumptions, demonstrating that with appropriate parameter selection, it achieves linear convergence to the true low-rank tensor at a constant rate, independent of the condition number. To enhance its practical applicability, we further propose a learnable self-supervised deep unfolding model, which enables effective parameter learning. Numerical experiments on both synthetic and real-world datasets demonstrate the superior performance of the proposed methods while maintaining competitive computational efficiency, especially consuming less time than RTPCA-TNN.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
