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On The Synergy Between Nonconvex Extensions of The Tensor Nuclear Norm for Tensor Recovery

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Document pages: 9 pages

Abstract: Low-rank tensor recovery has attracted much attention among various tensorrecovery approaches. A tensor rank has several definitions, unlike the matrixrank--e.g. the CP rank and the Tucker rank. Many low-rank tensor recoverymethods are focused on the Tucker rank. Since the Tucker rank is nonconvex anddiscontinuous, many relaxations of the Tucker rank have been proposed, e.g.,the tensor nuclear norm, weighted tensor nuclear norm, and weighted tensorSchatten-$p$ norm. In particular, the weighted tensor Schatten-p norm has twoparameters, the weight and $p$, and the tensor nuclear norm and weighted tensornuclear norm are special cases of these parameters. However, there has been nodetailed discussion of whether the effects of the weighting and $p$ aresynergistic. In this paper, we propose a novel low-rank tensor completion modelusing the weighted tensor Schatten-$p$ norm to reveal the relationships betweenthe weight and $p$. To clarify whether complex methods such as the weightedtensor Schatten-$p$ norm are necessary, we compare them with a simple methodusing rank-constrained minimization. It was found that the simple methods didnot outperform the complex methods unless the rank of the original tensor couldbe 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 fromobservations. These results are consistent with existing reports.

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