The best rank-one approximation ratio of a tensor space

Abstract

In this paper we define the best rank-one approximation ratio of a tensor space. It turns out that in the finite dimensional case this provides an upper bound for the quotient of the residual of the best rank-one approximation of any tensor in that tensor space and the norm of that tensor. This upper bound is strictly less than one, and it gives a convergence rate for the greedy rank-one update algorithm. For finite dimensional general tensor spaces, third order finite dimensional symmetric tensor spaces, and finite biquadratic tensor spaces, we give positive lower bounds for the best rank-one approximation ratio. For finite symmetric tensor spaces and finite dimensional biquadratic tensor spaces, we give upper bounds for this ratio.