  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.
