Multiscale interpolative construction of quantized tensor trains

TL;DR

This paper develops a multiscale polynomial interpolation approach to better understand and improve the construction of quantized tensor trains (QTTs), enabling efficient approximation of functions with various features.

Contribution

It introduces a multiscale interpolation perspective that explains QTT rank behavior and proposes new algorithms for QTT construction from function evaluations.

Findings

QTT ranks decay with increasing depth due to multiscale properties

Smoothness of the target function controls QTT rank

Functions with sharp features can still be effectively approximated by QTTs

Abstract

Quantized tensor trains (QTTs) have recently emerged as a framework for the numerical discretization of continuous functions, with the potential for widespread applications in numerical analysis. However, the theory of QTT approximation is not fully understood. In this work, we advance this theory from the point of view of multiscale polynomial interpolation. This perspective clarifies why QTT ranks decay with increasing depth, quantitatively controls QTT rank in terms of smoothness of the target function, and explains why certain functions with sharp features and poor quantitative smoothness can still be well approximated by QTTs. The perspective also motivates new practical and efficient algorithms for the construction of QTTs from function evaluations on multiresolution grids.