Single-entry computation of analytical hierarchical (binary) tree structures

Abstract

In this work, we present an algorithm that turns our analytical tree data-structures into accurate and efficient interpolation tools. First, our scheme provides an effortless and accurate approximation to the hierarchical Tucker decomposition (HTD) of a high-dimensional dense target tensor. We achieve this through low-dimensional polynomial fit of the (leaves) factor matrices. These reference factors can be obtained from the HTD of a much coarser tensor with the same number of modes and same domain of definition as the targeted one. Second, we provide a pass rule for the sample index, via the so-called chain-of-operators form, which avoids the calculation of the entire Tucker frame tree during the regression. We show that this single-entry based computational scheme leads to the embarrassingly parallel computation of the targeted tensor. To illustrate these results, we compare and discuss our results, in terms of CPU cost and storage, to the most commonly used tensor decomposition schemes and their associated algorithms.