Tensors, which give a faithful and effective representation to deliver the intrinsic structure of multi-dimensional data, play a crucial role in an increasing number of signal processing and machine learning problems. However, tensor data are often accompanied by arbitrary signal corruptions, including missing entries and sparse noise. A fundamental challenge is to reliably extract the meaningful information from corrupted tensor data in a statistically and computationally efficient manner. This paper develops a scaled gradient descent (ScaledGD) algorithm to directly estimate the tensor factors with tailored spectral initializations under the tensor-tensor product (t-product) and tensor singular value decomposition (t-SVD) framework. With tailored variants for tensor robust principal component analysis, (robust) tensor completion and tensor regression, we theoretically show that ScaledGD achieves linear convergence at a constant rate that is independent of the condition number of the ground truth low-rank tensor, while maintaining the low per-iteration cost of gradient descent. To the best of our knowledge, ScaledGD is the first algorithm that provably has such properties for low-rank tensor estimation with the t-SVD. Finally, numerical examples are provided to demonstrate the efficacy of ScaledGD in accelerating the convergence rate of ill-conditioned low-rank tensor estimation in a number of applications.

翻译：张量为描述多维数据内在结构提供了忠实且有效的表示，在日益增多的信号处理和机器学习问题中发挥着关键作用。然而，张量数据常伴随任意信号损坏，包括缺失项和稀疏噪声。一个根本性挑战在于如何以统计和计算高效的方式从损坏的张量数据中可靠地提取有意义的信息。本文在张量-张量积（t-product）和张量奇异值分解（t-SVD）框架下，开发了一种缩放梯度下降（ScaledGD）算法，通过定制化谱初始化直接估计张量因子。针对张量鲁棒主成分分析、（鲁棒）张量补全和张量回归等任务设计了定制化变体，我们从理论上证明ScaledGD能以恒定速率实现线性收敛，且该速率与真实低秩张量的条件数无关，同时保持了梯度下降的低单次迭代成本。据我们所知，ScaledGD是首个在t-SVD框架下被严格证明具有此类性质的低秩张量估计算法。最后，通过数值算例展示了ScaledGD在多种应用中加速病态低秩张量估计收敛速率的有效性。