On Low-Rank Hankel Matrix Denoising

TL;DR

This paper introduces a novel iterative algorithm for denoising low-rank Hankel matrices by combining structured low-rank approximation with data-driven singular value shrinkage, outperforming existing methods.

Contribution

It proposes a new approach that integrates structured low-rank approximation with singular value shrinkage for improved Hankel matrix denoising.

Findings

The proposed method outperforms existing algorithms in noise-free matrix estimation.

Numerical results demonstrate superior performance in trajectory and impulse response denoising.

The approach effectively leverages structured low-rank properties for denoising tasks.

Abstract

The low-complexity assumption in linear systems can often be expressed as rank deficiency in data matrices with generalized Hankel structure. This makes it possible to denoise the data by estimating the underlying structured low-rank matrix. However, standard low-rank approximation approaches are not guaranteed to perform well in estimating the noise-free matrix. In this paper, recent results in matrix denoising by singular value shrinkage are reviewed. A novel approach is proposed to solve the low-rank Hankel matrix denoising problem by using an iterative algorithm in structured low-rank approximation modified with data-driven singular value shrinkage. It is shown numerically in both the input-output trajectory denoising and the impulse response denoising problems, that the proposed method performs the best in terms of estimating the noise-free matrix among existing algorithms of low-rank matrix approximation and denoising.