A Review on Adaptive Low-Rank Approximation Techniques in the Hierarchical Tensor Format

Abstract

The hierarchical tensor format allows for the low-parametric representation of tensors even in high dimensions d. On the one hand, this format provides a robust framework for approximate arithmetic operations with tensors based on rank truncations, which can be exploited in iterative algorithms. On the other hand, it can be used for the direct approximation of high-dimensional data stemming, e.g., from the discretisation of multivariate functions. In this review, we discuss several strategies for an adaptive approximation of tensors in the hierarchical format by black box type techniques, including problems of tensor reconstruction and tensor completion.