Computationally efficient selection criterion of initial equation in STLN-based structured low-rank approximation

Abstract

The problem of structured low-rank approximation (SLRA) of a given matrix arises in a wide range of applications. This problem can be solved by the structured total least norm (STLN) method, which formulates it as a linear equation called an “initial equation”. Because this equation directly impacts the performance of STLN-based SLRA, certain selection criteria have been proposed. However, these criteria are computationally intensive and their theoretical validity has not been sufficiently investigated. In this paper, we theoretically analyze the relationship between the original problem and an initial equation, and subsequently propose a novel selection criterion with a computational order of $O\left(n^3\right)$, in contrast to the complexity of $O\left(n^4\right)$ incurred by conventional methods. We also demonstrate numerical experiments for Sylvester matrices to confirm that our selection criterion reduces not only the selection time of the initial equation, but also the overall calculation time of the STLN-based SLRA.