The nuclear norm and Schatten-$p$ quasi-norm are popular rank proxies in low-rank matrix recovery. However, computing the nuclear norm or Schatten-$p$ quasi-norm of a tensor is hard in both theory and practice, hindering their application to low-rank tensor completion (LRTC) and tensor robust principal component analysis (TRPCA). In this paper, we propose a new class of tensor rank regularizers based on the Euclidean norms of the CP component vectors of a tensor and show that these regularizers are monotonic transformations of tensor Schatten-$p$ quasi-norm. This connection enables us to minimize the Schatten-$p$ quasi-norm in LRTC and TRPCA implicitly via the component vectors. The method scales to big tensors and provides an arbitrarily sharper rank proxy for low-rank tensor recovery compared to the nuclear norm. On the other hand, we study the generalization abilities of LRTC with the Schatten-$p$ quasi-norm regularizer and LRTC with the proposed regularizers. The theorems show that a relatively sharper regularizer leads to a tighter error bound, which is consistent with our numerical results. Particularly, we prove that for LRTC with Schatten-$p$ quasi-norm regularizer on $d$-order tensors, $p=1/d$ is always better than any $p>1/d$ in terms of the generalization ability. We also provide a recovery error bound to verify the usefulness of small $p$ in the Schatten-$p$ quasi-norm for TRPCA. Numerical results on synthetic data and real data demonstrate the effectiveness of the regularization methods and theorems.

翻译：核范数和Schatten-p拟范数是低秩矩阵恢复中常用的秩代理函数。然而，在理论与实践中计算张量的核范数或Schatten-p拟范数均存在困难，这限制了它们在低秩张量补全（LRTC）和张量鲁棒主成分分析（TRPCA）中的应用。本文提出了一类基于张量CP分量向量欧几里得范数的新型张量秩正则化器，并证明了这些正则化器是张量Schatten-p拟范数的单调变换。这一联系使我们能够通过分量向量隐式地最小化LRTC和TRPCA中的Schatten-p拟范数。该方法可扩展至大规模张量，并提供比核范数更尖锐的低秩张量恢复秩代理函数。另一方面，我们研究了采用Schatten-p拟范数正则化器的LRTC与所提正则化器LRTC的泛化能力。理论表明，相对更尖锐的正则化器能产生更紧致的误差界，这与数值结果一致。特别地，我们证明对于d阶张量上的Schatten-p拟范数正则化LRTC，在泛化能力方面p=1/d始终优于任何p>1/d。我们还提供了恢复误差界，以验证TRPCA中Schatten-p拟范数取小p值的有效性。合成数据与真实数据的数值结果验证了所提正则化方法及理论的有效性。