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The nuclear norm and Schatten-$p$ quasi-norm are popular rank proxies inlow-rank matrix recovery. However, computing the nuclear norm or Schatten-$p$quasi-norm of a tensor is hard in both theory and practice, hindering theirapplication to low-rank tensor completion (LRTC) and tensor robust principalcomponent analysis (TRPCA). In this paper, we propose a new class of tensorrank regularizers based on the Euclidean norms of the CP component vectors of atensor and show that these regularizers are monotonic transformations of tensorSchatten-$p$ quasi-norm. This connection enables us to minimize theSchatten-$p$ quasi-norm in LRTC and TRPCA implicitly via the component vectors.The method scales to big tensors and provides an arbitrarily sharper rank proxyfor 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-normregularizer and LRTC with the proposed regularizers. The theorems show that arelatively sharper regularizer leads to a tighter error bound, which isconsistent with our numerical results. Particularly, we prove that for LRTCwith Schatten-$p$ quasi-norm regularizer on $d$-order tensors, $p=1/d$ isalways better than any $p>1/d$ in terms of the generalization ability. We alsoprovide a recovery error bound to verify the usefulness of small $p$ in theSchatten-$p$ quasi-norm for TRPCA. Numerical results on synthetic data and realdata demonstrate the effectiveness of the regularization methods and theorems.

Euclidean-Norm-Induced Schatten-p Quasi-Norm Regularization for Low-Rank Tensor Completion and Tensor Robust Principal Component Analysis