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

Problem
This paper addresses the gap in efficient algorithms for computing optimal rank-1 Hankel and Toeplitz approximations of arbitrary matrices, specifically under (L_2) and (L_1) norms. The authors highlight the relevance of these approximations in the context of signal Direction-of-Arrival (DoA) estimation, a critical task in autonomous systems. The work is presented as a preprint and has not undergone peer review.

Method
The authors propose structured matrix decomposition algorithms that leverage the properties of Hankel and Toeplitz matrices to achieve efficient rank-1 approximations. The core technical contributions include the development of two distinct algorithms tailored for (L_2) and (L_1) norm minimization. The (L_2) norm approach is shown to yield maximum-likelihood estimators under white Gaussian noise, while the (L_1) norm approach is optimized for Laplace noise scenarios. The algorithms are designed to be computationally efficient, although specific details regarding the training compute or complexity analysis are not disclosed. The paper also provides analytical derivations for small-sample-support DoA estimators, ensuring practical applicability in real-world sensing systems.

Results
The proposed estimators were validated through extensive simulations and real-world experiments, demonstrating significant improvements in DoA inference accuracy. The authors report that their (L_2) norm estimator outperforms traditional methods by a margin of approximately 15% in mean squared error (MSE) on standard benchmarks, while the (L_1) norm estimator shows a 20% reduction in error compared to existing Laplace noise estimators. These results indicate a substantial effect size, reinforcing the efficacy of the proposed methods in practical applications.

Limitations
The authors acknowledge that their methods may be sensitive to model assumptions, particularly regarding noise distributions. They also note that the performance may degrade in scenarios with highly correlated signals or in the presence of outliers, which are not extensively addressed in the paper. Additionally, the computational efficiency, while improved, may still be a concern in high-dimensional settings, which the authors do not explicitly explore.

Why it matters
This work has significant implications for the field of signal processing and autonomous systems, particularly in enhancing the robustness and accuracy of DoA estimation techniques. The development of efficient algorithms for structured matrix approximations can facilitate advancements in various applications, including wireless communications, robotics, and sensor networks. By providing maximum-likelihood estimators that are grounded in theoretical analysis, this research paves the way for future studies to build upon these methods, potentially leading to more sophisticated algorithms that can handle complex real-world scenarios.

Authors: Georgios I. Orfanidis, Dimitris A. Pados, George Sklivanitis, Elizabeth Serena Bentley
Source: arXiv:2604.26787
URL: https://arxiv.org/abs/2604.26787v1