Biquadratic Tensors, Biquadratic Decomposition and Norms of Biquadratic Tensors

TL;DR

This paper investigates the properties, decompositions, and norms of biquadratic tensors, providing methods to compute spectral and nuclear norms efficiently, establishing existence of rank-one decompositions, and analyzing bounds related to invertible tensors.

Contribution

It introduces new methods for computing norms of biquadratic tensors, proves the existence of biquadratic rank-one decompositions, and analyzes bounds for invertible biquadratic tensors.

Findings

Spectral and nuclear norms can be computed using biquadratic structure.

A biquadratic rank-one decomposition always exists for such tensors.

Bounds are established for the nuclear norm and products involving invertible biquadratic tensors.

Abstract

Biquadratic tensors play a central role in many areas of science. Examples include elasticity tensor and Eshelby tensor in solid mechanics, and Riemann curvature tensor in relativity theory. The singular values and spectral norm of a general third order tensor are the square roots of the M-eigenvalues and spectral norm of a biquadratic tensor. The tensor product operation is closed for biquadratic tensors. All of these motivate us to study biquadratic tensors, biquadratic decomposition and norms of biquadratic tensors. We show that the spectral norm and nuclear norm for a biquadratic tensor may be computed by using its biquadratic structure. Then, either the number of variables is reduced, or the feasible region can be reduced. We show constructively that for a biquadratic tensor, a biquadratic rank-one decomposition always exists, and show that the biquadratic rank of a biquadratic tensor is preserved under an independent biquadratic Tucker decomposition. We present a lower bound and an upper bound of the nuclear norm of a biquadratic tensor. Finally, we define invertible biquadratic tensors, and present a lower bound for the product of the nuclear norms of an invertible biquadratic tensor and its inverse, and a lower bound for the product of the nuclear norm of an invertible biquadratic tensor, and the spectral norm of its inverse.