r-Term Approximation

Abstract

In general, one tries to approximate a tensor v by another tensor u requiring less data. The reason is twofold: the memory size should decrease and, hopefully, operations involving u should require less computational work. In fact, ${\rm u} \in \mathcal{R}_{{\rm r}}$ leads to decreasing cost for storage and operations as r decreases. However, the other side of the coin is an increasing approximation error. Correspondingly, in Sect. 9.1 two approximation strategies are presented, where either the representation rank r of u or the accuracy is prescribed. Before we study the approximation problem in general, two particular situations are discussed. Section 9.2 is devoted to r = 1, when ${\rm u} \in \mathcal{R}_{1}$ is an elementary tensor. The matrix case d = 2 is recalled in Sect. 9.3. The properties observed in the latter two sections contrast with the true tensor case studied in Sect. 9.4. Numerical algorithms solving the approximation problem will be discussed in Sect. 9.5. Modified approximation problems are addressed in Sect. 9.6.