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

2026-04-29

AI Summary

This study addresses the problem of computing the best rank-1 Hankel or Toeplitz approximation of an arbitrary matrix under both L₂ and L₁ norms, and applies this framework to small-sample direction-of-arrival (DoA) estimation. To this end, the authors propose an efficient and exact structured matrix factorization algorithm and derive maximum-likelihood-optimal DoA estimators under Gaussian and Laplacian noise assumptions, respectively. By integrating structured matrix approximation, norm-based optimization, and signal processing techniques, the proposed method achieves superior estimation accuracy and robustness in limited-data regimes. Extensive simulations and real-world experiments demonstrate its significant performance gains over existing approaches under low-snapshot conditions.

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.

Problem

• Hankel matrix
• Toeplitz matrix
• rank-1 approximation
• Direction-of-Arrival estimation
• structured matrix approximation

Innovation

• Rank-1 decomposition
• Hankel matrix
• Toeplitz matrix
• Direction-of-Arrival estimation
• Few-shot inference