Tensor spectral \({\textbf{p}}\) -norms are generalizations of matrix induced norms. Matrix induced norms are an important type of matrix norms, and tensor spectral \({\textbf{p}}\) -norms are also important in applications. We discuss some basic properties of tensor spectral \({\textbf{p}}\) -norms. We extend the submultiplicativity of the matrix spectral 2-norm to the tensor case, based on which we give a bound of the tensor spectral 2-norm and provide a fast method for computing spectral 2-norms of sum-of-squares tensors. To compute tensor spectral \({\textbf{p}}\) -norms, we propose a higher order power method. Experiments show the high efficiency of the proposed methods and numerical results on spectral \({\textbf{p}}\) -norms of random tensors are also given.
