Tensor rank bounds and explicit QTT representations for the inverses of circulant matrices

TL;DR

This paper investigates the inversion of circulant matrices and their representation in the quantized tensor-train (QTT) format, providing bounds on QTT ranks and explicit representations under certain conditions, aiding numerical solutions.

Contribution

It establishes bounds on QTT ranks for inverses of circulant matrices and derives explicit QTT representations under specific assumptions, advancing tensor-based numerical methods.

Findings

Inverse circulant matrices admit QTT representations with ranks bounded by (m+n).

Explicit QTT representations are derived under certain conditions on matrix entries.

The results facilitate stable numerical solutions of differential equations with periodic boundary conditions.

Abstract

In this paper, we are concerned with the inversion of circulant matrices and their quantized tensor-train (QTT) structure. In particular, we show that the inverse of a complex circulant matrix $A$, generated by the first column of the form $(a_0,\dots,a_{m-1},0,\dots,0,a_{-n},\dots, a_{-1})^\top$ admits a QTT representation with the QTT ranks bounded by $(m+n)$. Under certain assumptions on the entries of $A$, we also derive an explicit QTT representation of $A^{-1}$. The latter can be used, for instance, to overcome stability issues arising when numerically solving differential equations with periodic boundary conditions in the QTT format.