  Low-rank tensor recovery has attracted much attention among various tensor
recovery approaches. A tensor rank has several definitions, unlike the matrix
rank--e.g. the CP rank and the Tucker rank. Many low-rank tensor recovery
methods are focused on the Tucker rank. Since the Tucker rank is nonconvex and
discontinuous, many relaxations of the Tucker rank have been proposed, e.g.,
the tensor nuclear norm, weighted tensor nuclear norm, and weighted tensor
Schatten-$p$ norm. In particular, the weighted tensor Schatten-p norm has two
parameters, the weight and $p$, and the tensor nuclear norm and weighted tensor
nuclear norm are special cases of these parameters. However, there has been no
detailed discussion of whether the effects of the weighting and $p$ are
synergistic. In this paper, we propose a novel low-rank tensor completion model
using the weighted tensor Schatten-$p$ norm to reveal the relationships between
the weight and $p$. To clarify whether complex methods such as the weighted
tensor Schatten-$p$ norm are necessary, we compare them with a simple method
using rank-constrained minimization. It was found that the simple methods did
not outperform the complex methods unless the rank of the original tensor could
be accurately known. If we can obtain the ideal weight, $p = 1$ is sufficient,
although it is necessary to set $p<1$ when using the weights obtained from
observations. These results are consistent with existing reports.
