2019-10-14 | Liqun Qi：Nuclear Norm and Spectral Norm of Tensor Product
We show that the nuclear norm of the tensor product of two tensors is not greater than the product of the nuclear norms of these two tensors. As an application, we give lower bounds for the nuclear norm of an arbitrary tensor. We show that the spectral norm of the tensor product of two tensors is not greater than the product of the spectral norm of one tensor, and the nuclear norm of another tensor. By this result, we present an alternative formula for the spectral norm of a tensor, and give some lower bounds for the product of the nuclear norm and the spectral norm of an arbitrary tensor. The first result also shows that the nuclear norm of the square matrix is a matrix norm. We then extend the concept of matrix norm to tensor norm. A real function defined for all real tensors is called a tensor norm if it is a norm for any tensor space with fixed dimensions, and the norm of the tensor product of two tensors is always not greater than the product of the norms of these two tensors. We show that the $1$-norm, the Frobenius norm and the nuclear norm of tensors are tensor norms but the infinity norm and the spectral norm of tensors are not tensor norms.
Liqun Qi is now Professor of Applied Mathematics and Head of Department of Applied Mathematics at the Hong Kong Polytechnic University. Professor Qi has published more than 200 research papers in international journals. He established the superlinear and quadratic convergence theory of the semismooth Newton method, and played a principal role in the development of reformulation methods in optimization. Professor Qi's research work has been cited by the researchers around the world. According to the authoritative citation database www.isihighlycited.com, he is one of the world's most highly cited 300 mathematicians. In 2005, Professor Qi pioneered the research on eigenvalues for higher order tensors, which now has applications in biomedical engineering, statistical data analysis, spectral hypergraph theory, solid mechanics, etc. In 2010, Professor Qi received the First Class Science and Technology Award of Chinese Operations Research Society.