Yet Another Schatten Norm for Tensor Recovery

In this paper, we introduce a new class of Schatten norms for tensor recovery. In the new norm, unfoldings of a tensor along not only every single order but also all combinations of orders are taken into account. Additionally, we prove that the proposed tensor norm has similar properties to matrix Schatten norm, and also provides several propositions which is useful in the recovery problem. Furthermore, for reliable recovery of a tensor with Gaussian measurements, we show the necessary size of measurements using the new norm. Compared to using conventional overlapped Schatten norm, the new norm results in less measurements for reliable recovery with high probability. Finally, experimental results demonstrate the efficiency of the new norm in video in-painting.