We study the decomposability and the subdifferential of the tensor nuclear norm. Both concepts are well understood and widely applied in matrices but remain unclear for higher-order tensors. We show that the tensor nuclear norm admits a full decomposability over specific subspaces and determine the largest possible subspaces that allow the full decomposability. We derive novel inclusions of the subdifferential of the tensor nuclear norm and study its subgradients in a variety of subspaces of interest. All the results hold for tensors of an arbitrary order. As an immediate application, we establish the statistical performance of the tensor robust principal component analysis, the first such result for tensors of an arbitrary order.

翻译：本文研究张量核范数的可分解性与次微分性质。这两个概念在矩阵理论中已得到充分理解并广泛应用，但对于高阶张量仍不明确。我们证明张量核范数在特定子空间上具有完全可分解性，并确定了允许完全可分解的最大子空间。我们推导了张量核范数次微分的新型包含关系，并在多个重要子空间中研究了其次梯度。所有结论适用于任意阶张量。作为直接应用，我们建立了张量鲁棒主成分分析的统计性能，这是针对任意阶张量的首个此类结果。