Multivariate Cross Approximation

Abstract

An important feature is the computation of a tensor from comparably few tensor entries. The input tensor v 2 V is assumed to be given in a full functional representation so that, on request, any entry can be determined. This partial information can be used to determine an approximation ~v 2 V. In the matrix case (d=2) an algorithm for this purpose is well-known under the name ‘cross approximation’ or ‘adaptive cross approximation’ (ACA). The generalisation to the multi-dimensional case is not straightforward. We present a multivariate cross approximation, which fits to the hierarchical format. If v 2 Hr holds with a known rank vector r, this tensor can be reproduced exactly, i.e., ~v = v. In the general case, the approximation is heuristic. Exact error estimates require either inspection of all tensor entries (which is practically impossible) or strong theoretical a priori knowledge.