Tree Adaptive Approximation in the Hierarchical Tensor Format

Abstract

The hierarchical tensor format allows for the low-parametric representation of tensors even in high dimensions $d$. The efficiency of this representation strongly relies on an appropriate hierarchical splitting of the different directions $1,\ldots,d$ such that the associated ranks remain sufficiently small. This splitting can be represented by a binary tree which is usually assumed to be given. In this paper, we address the question of finding an appropriate tree from a subset of tensor entries without any a priori knowledge on the tree structure. We propose an agglomerative strategy that can be combined with rank-adaptive cross approximation techniques such that tensors can be approximated in the hierarchical format in an entirely black box way. Numerical examples illustrate the potential and the limitations of our approach.