Estimates for the quantized tensor train ranks for the power functions

TL;DR

The paper addresses estimating quantized tensor train (QTT) ranks for the power functions $f(k)=k^{-\\alpha}$ with $\\alpha>1$ sampled at $k=1,\dots,2^{d}$. It develops rank estimates by connecting QTT unfoldings to a Hankel matrix $H(d,f)$ and invoking the Hamburger moment problem, yielding a TT-rank bound that grows like $C d(\ln d + \ln(1/\varepsilon) + \ln M)$. The authors validate the theory numerically with the TT-SVD-based approach, demonstrating small QTT ranks even for large $d$ and a logarithmic dependence on the approximation tolerance $\varepsilon$. These results justify the use of QTT representations to store and compute solutions in aggregation-fragmentation equations, enabling scalable simulations in high dimensions.

Abstract

In this work we provide theoretical estimates for the ranks of the power functions $f(k) = k^{-α}$, $α>1$ in the quantized tensor train (QTT) format for $k = 1, 2, 3, \ldots, 2^{d}$. Such functions and their several generalizations (e.~g. $f(k) = k^{-α} \cdot e^{-λk}, λ> 0$) play an important role in studies of the asymptotic solutions of the aggregation-fragmentation kinetic equations. In order to support the constructed theory we verify the values of QTT-ranks of these functions in practice with the use of the TTSVD procedure and show an agreement between the numerical and analytical results.

Figures (1)

• Figure 1: QTT approximations with multiple tolerance levels for $d=22$ and $\alpha=1.5, 2.5$. Approximation with TTSVD accuracy parameter $\varepsilon=10^{-9}$ requires just maximal QTT-rank $R=7$, each increase of this rank by 1 allows to increase the accuracy of the approximation by dozens of times.

Theorems & Definitions (8)

• Proposition 1
• proof
• Proposition 2
• proof
• Theorem 1
• proof
• Remark 1
• Remark 2