Robust Low-Rank Tensor Decomposition with the L₂ Criterion

The growing prevalence of tensor data, or multiway arrays, in science and engineering applications motivates the need for tensor decompositions that are robust against outliers. In this article, we present a robust Tucker decomposition estimator based on the L₂ criterion, called the Tucker-L2E. Our numerical experiments demonstrate that Tucker-L2E has empirically stronger recovery performance in more challenging high-rank scenarios compared with existing alternatives. The appropriate Tucker-rank can be selected in a data-driven manner with cross-validation or hold-out validation. The practical effectiveness of Tucker-L2E is validated on real data applications in fMRI tensor denoising, PARAFAC analysis of fluorescence data, and feature extraction for classification of corrupted images.张量数据（tensor data）或称多维数组（multiway arrays）在科研与工程领域的应用日益普及，这催生了对可抵御异常值的鲁棒张量分解方法的需求。本文提出了一种基于L₂准则的鲁棒塔克（Tucker）分解估计器，命名为Tucker-L2E。我们的数值实验表明，相较于现有同类方法，Tucker-L2E在更具挑战性的高秩场景下展现出更优异的经验恢复性能。可通过交叉验证或留出验证（hold-out validation）以数据驱动的方式选取合适的塔克秩（Tucker-rank）。Tucker-L2E的实际有效性已在功能磁共振成像（fMRI）张量去噪、荧光数据的并行因子（PARAFAC）分析以及受损图像分类的特征提取等真实数据应用中得到验证。