Shengke Xue

Fig. 5 Illustration of the t-SVD of an n1×n2×n3n1×n2×n3 tensor, i.e., A=U∗S∗VTA=U∗S∗VT. Our tensor nuclear norm ∥A∥∗‖A‖∗ is defined as the sum of singular values of all frontal slices of the f-diagonal SS, i.e., ∥A∥∗≜tr(S)=∑n3i=1tr(S(i))=tr(S¯(1))=∥A¯(1)∥∗‖A‖∗≜tr(S)=∑i=1n3tr(S(i))=tr(S¯(1))=‖A¯(1)‖∗. Note that our tensor nuclear norm becomes standard matrix nuclear norm when n3=1n3=1. Thus, our tensor nuclear norm can be considered as a direct extension from the matrix case to the tensor case.