Approximation by tree tensor networks in high dimensions: Sobolev and compositional functions

TL;DR

This paper develops new convergence estimates for approximating high-dimensional functions using tree tensor networks, especially focusing on Sobolev and compositional functions, demonstrating their efficiency without the curse of dimensionality.

Contribution

It introduces refined approximation estimates for high-dimensional functions with Sobolev and compositional structures using tree tensor networks, including a constructive encoding method.

Findings

Tree tensor networks can approximate compositional functions without curse of dimensionality.

New convergence estimates are derived for Sobolev and mixed Sobolev functions.

Approximation efficiency depends on the depth of the underlying tree structure.

Abstract

This paper is concerned with convergence estimates for fully discrete tree tensor network approximations of high-dimensional functions from several model classes. For functions having standard or mixed Sobolev regularity, new estimates generalizing and refining known results are obtained, based on notions of linear widths of multivariate functions. In the main results of this paper, such techniques are applied to classes of functions with compositional structure, which are known to be particularly suitable for approximation by deep neural networks. As shown here, such functions can also be approximated by tree tensor networks without a curse of dimensionality -- however, subject to certain conditions, in particular on the depth of the underlying tree. In addition, a constructive encoding of compositional functions in tree tensor networks is given.