Hankel and Toeplitz Rank-1 Decomposition of Arbitrary Matrices with Applications to Signal Direction-of-Arrival Estimation

TL;DR

This paper introduces efficient algorithms for rank-1 Hankel and Toeplitz matrix approximations, enhancing signal direction-of-arrival estimation accuracy in autonomous systems.

Contribution

It develops novel structured matrix decomposition algorithms and analytically grounded estimators for practical DoA applications, optimized under different noise models.

Findings

Estimators are maximum-likelihood optimal under Gaussian and Laplace noise.

Algorithms are accurate and computationally efficient.

Validated through simulations and real-world data experiments.

Abstract

We consider the problems of computing the optimal rank-$1$ Hankel and Toeplitz-structured approximation of arbitrary matrices under $L_2$ and $L_1$-norm error. Such problems arise naturally in engineered systems, including the basic few-shot signal Direction-of-Arrival (DoA) estimation problem that is of importance to modern autonomous systems applications. We develop accurate and computationally efficient structured matrix decomposition algorithms for both formulations and then derive analytically grounded small-sample-support DoA estimators for practical sensing system deployments. The resulting estimators under the $L_2$ and $L_1$ norms are formally shown to be maximum-likelihood optimal under white Gaussian and Laplace noise, respectively. The estimators are further validated through extensive simulation studies and real-world data experiments in few-shot DoA inference.