Multiscale interpolative construction of quantized tensor trains

TL;DR

The paper addresses the gap in understanding how quantized tensor trains (QTTs) approximate functions by reframing QTT as a multiscale polynomial interpolation problem. It derives depth-aware TT-rank bounds, notably showing that $\,r_m^{(\varepsilon,\infty)}[T]$ can decay with depth and that $\,\Omega$-bandlimited functions admit ranks near $\sqrt{\Omega}$, improving over naive exponential-exponential estimates. The authors introduce practical interpolative QTT constructions (basic, rank-revealing, sparse, and a priori decaying-rank variants), along with a method to invert the construction and recover grid evaluations, and extend the approach to multiresolution and multivariate settings with interleaved and serial orderings. Extensive numerical experiments corroborate the theory, showing strong performance and robustness against standard tensor cross interpolation, while enabling efficient hybrid QTT/multiresolution workflows. Overall, the work clarifies the link between multiscale interpolation, spectral content, and TT representability, and provides actionable algorithms for constructing and manipulating QTTs from function evaluations across multiple scales and dimensions.

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.

Figures (14)

• Figure 2.1: Tensor network diagram for MPS/TT (\ref{['eq:tt']}). The nodes indicate tensor cores. Shared edges indicate contracted indices, while open edges indicate indices of the tensor $T(\sigma_{1},\ldots,\sigma_{6})$ represented by the diagram.
• Figure 4.1: The core $A$ interpolates values from the Chebyshev-Lobatto grid $\{x_{\leq m}+2^{-m}c^{\alpha}\}_{\alpha=0}^{N}$ for the dyadic subinterval $[x_{\leq m},x_{\leq m}+2^{-m}]$, depicted on top, to the Chebyshev-Lobatto grids $\{x_{\leq m}+2^{-(m+1)}\sigma+2^{-(m+1)}c^{\beta}\}_{\beta=0}^{N}$ for the left $(\sigma=0$) and right $(\sigma=1$) halves of this dyadic subinterval. The interpolation is indicated graphically in the figure for the case $\sigma=0$.
• Figure 4.2: Top: tensor network diagram for the tensor $A^{p}$, where $p=4$. Bottom: the full QTT $A_{\mathrm{L}}A^{K-2}A_{\mathrm{R}}$, where $K=6$.
• Figure 4.3: Illustration of the rank-revealing interpolative construction of QTT.
• Figure 5.1: Top: stage 1 of the inversion procedure. Bottom: one level of stage 2.

Theorems & Definitions (29)

• Lemma 1
• proof
• Proposition 2
• Remark 3
• Definition 4
• Proposition 5
• Remark 6
• Definition 7
• Proposition 8
• Remark 9