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, u 2 Rr 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 Section 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, i.e., u 2 R1 is an elementary tensor. The matrix case d = 2 is recalled in Section 9.3. The properties observed in the latter two cases contrast with the true tensor case studied in Section 9.4. The numerical difficulties are caused by the fact that the r-term format is not closed. In Section 9.5 we study nonclosed formats in general. Numerical algorithms, in particular the ALS method, solving the approximation problem will be discussed in Section 9.6. Modified approximation problems are addressed in Section 9.7. Section 9.8 provides important r-term approximations for special functions and operators. It is shown that the Coulomb potential 1=jx yj as well as the inverse of the Laplace operator allow r-term approximations with exponentially improving accuracy.