Approximation Theory of Tree Tensor Networks: Tensorized Multivariate Functions

TL;DR

This paper investigates the approximation capabilities of tensor networks for multivariate functions, demonstrating their near-optimal expressivity across various smoothness classes and their ability to approximate functions beyond classical smoothness spaces.

Contribution

It provides theoretical insights into the approximation power of tensor networks, showing their universal expressivity and embedding properties relative to classical smoothness spaces.

Findings

TNs can nearly optimally replicate spline approximations for any smoothness.

Tensor networks exhibit universal expressivity comparable to deep ReLU networks.

TNs can approximate functions outside classical smoothness spaces.

Abstract

We study the approximation of multivariate functions with tensor networks (TNs), providing some answers to the following two questions: ``what are the approximation capabilities of TNs for functions from classical smoothness classes?'' and ``what are the properties of the class of functions that can be approximated with TNs with a certain performance?'' As a partial answer to the former, we show that TNs can (near to) optimally replicate $h$-uniform and $h$-adaptive spline approximation, for any smoothness order of the target function. Tensor networks thus exhibit universal expressivity w.r.t. isotropic, anisotropic and mixed smoothness spaces that is comparable with more general neural networks families such as deep rectified linear unit (ReLU) networks. Put differently, TNs have the capacity to (near to) optimally approximate many function classes -- without being adapted to the particular class in question. As a partial answer to the latter, as a candidate model class we consider approximation classes of TNs and show that these are (quasi-)Banach spaces, that many types of classical smoothness spaces are continuously embedded into said approximation classes and that TNs approximation classes are themselves not embedded in any classical smoothness space. In other words, TNs can efficiently approximate functions that lie beyond classical smoothness spaces.