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Nuclear Norm and Spectral Norm of Tensor Product
时间  Datetime
2019-10-11 10:00 — 11:00 
地点  Venue
639
报告人  Speaker
Liqun Qi
单位  Affiliation
The Hong Kong Polytechnic University
邀请人  Host
Jinyan Fan
报告摘要  Abstract

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.