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, . . . , 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.

Optimization on the Hierarchical Tucker manifold – Applications to tensor completion

“…Although this approach is limited to considerations of seismic data, for larger dimensions/different domains, potentially the method of [5] can choose an appropriate dimension tree automatically. In the next section, we also include the case when Ω ⊂ [n src ] × [n src ] × [n rec ] × [n rec ], i.e.…”

Hierarchical Tensor Approximation of Output Quantities of Parameter-Dependent PDEs

“…For some applications the choice of the tree might be straight-forward, or even irrelevant: if the tensor can be approximated in the CP-format with rank K, then any dimension tree is sufficient and the nodewise ranks can be bounded by K. If this is not the case, then a heuristic approach is required to find a useful tree. In [3] we present an agglomerative procedure to obtain the tree T D . There we also specify how the nodewise ranks can be estimated by sampling of random submatrices.…”

Efficient tree decomposition of high-rank tensors

“…Finally, for comparison purposes consider Example 6.4 of Ballani and Grasedyck [17]. Let Ising N -point functions for a system formed of a 3 × 3 square lattice with open boundary conditions.…”

“…This example is non-trivial because neither the structure of g nor the bitwise indexing of ξ permit any obvious factoring. Table I shows the effective dimension defined in Ballani and Grasedyck [17] for both their adaptive agglomeration scheme and the greedy one introduced in this work. To match the parameters used in their tests the dimension was set to N = 8, the tolerance = 10 −8 , and the test was performed for each α ∈ {0, 0.25, 0.5, 0.75, 1.0}.…”

“…For instance, if there is a physical reason to expect two sets of variables to be weakly coupled, they can be placed on opposing sides of the tree. More recently methods which automate the selection have been proposed [17] and found to perform well. The purpose of this work is to provide two new algorithms in this vein.…”