Zhang Xiaoqin, Wang Di, Zhou Zhengyuan, Ma Yi
IEEE Trans Pattern Anal Mach Intell. 2021 Jan;43(1):238-255. doi: 10.1109/TPAMI.2019.2929043. Epub 2020 Dec 4.
Low-rank tensor recovery in the presence of sparse but arbitrary errors is an important problem with many practical applications. In this work, we propose a general framework that recovers low-rank tensors, in which the data can be deformed by some unknown transformations and corrupted by arbitrary sparse errors. We give a unified presentation of the surrogate-based formulations that incorporate the features of rectification and alignment simultaneously, and establish worst-case error bounds of the recovered tensor. In this context, the state-of-the-art methods 'RASL' and 'TILT' can be viewed as two special cases of our work, and yet each only performs part of the function of our method. Subsequently, we study the optimization aspects of the problem in detail by deriving two algorithms, one based on the alternating direction method of multipliers (ADMM) and the other based on proximal gradient. We provide convergence guarantees for the latter algorithm, and demonstrate the performance of the former through in-depth simulations. Finally, we present extensive experimental results on public datasets to demonstrate the effectiveness and efficiency of the proposed framework and algorithms.
在存在稀疏但任意误差的情况下进行低秩张量恢复是一个具有许多实际应用的重要问题。在这项工作中，我们提出了一个恢复低秩张量的通用框架，其中数据可能会因一些未知变换而变形，并受到任意稀疏误差的破坏。我们对基于替代的公式进行了统一的阐述，这些公式同时纳入了校正和对齐的特征，并建立了恢复张量的最坏情况误差界。在此背景下，最先进的方法“RASL”和“TILT”可被视为我们工作的两个特殊情况，但它们各自仅执行了我们方法的部分功能。随后，我们通过推导两种算法详细研究了该问题的优化方面，一种基于乘子交替方向法（ADMM），另一种基于近端梯度。我们为后一种算法提供了收敛保证，并通过深入的模拟展示了前一种算法的性能。最后，我们在公共数据集上展示了广泛的实验结果，以证明所提出框架和算法的有效性和效率。
