  Based upon the T-SVD (tensor SVD) of third order tensors, introduced by
Kilmer and her collaborators, we define T-singular values of third order
tensors. T-singular values of third order tensors are nonnegative scalars. The
number of nonzero T-singular values is the tensor tubal rank of the tensor. We
then use T-singular values to define the tail energy of a third order tensor,
and apply it to the error estimation of a tensor sketching algorithm for low
rank tensor approximation. Numerical experiments on real world data show that
our algorithm is efficient.
