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

Motivated by sensing modalities in modern autonomous systems that involve hardware-constrained spatial sampling over large arrays with limited coherence time, we develop a novel framework for rapid super-resolution multi-signal direction-of-arrival (DoA) estimation based on Hankel-structured sensing and data matrix decomposition of arbitrary rank, under both the $L_2$ and $L_1$-norm formulation. The resulting $L_2$-norm estimator is shown to be maximum-likelihood optimal in white Gaussian noise. The $L_1$-norm estimator is shown to be maximum-likelihood optimal in independent, identically distributed (i.i.d.) isotropic Laplace noise, offering broad robustness to impulsive interference and corrupted measurements commonly encountered in practice. Extensive simulations demonstrate that the proposed methods exhibit powerful super-resolution capabilities, requiring significantly lower SNR and achieving substantially higher resolution probability than recent competing approaches.

Direction-of-arrival (DOA) estimation is an important task in microphone array processing and many downstream applications. The steered response power with phase transform (SRP-PHAT) method has been widely adopted for DOA estimation in recent years. However, accurate SRP-PHAT estimation in 3D scenarios requires evaluating steering responses over thousands of candidate directions, severely limiting real-time performance on resource-constrained platforms. This challenge becomes even more critical for planar arrays, which are widely used in robotics due to their structural simplicity. Motivated by the fact that azimuth estimation is usually more reliable than elevation estimation for most arrays, we propose ASAP, an azimuth-priority strip-based search approach to planar microphone array DOA estimation in 3D. In the first stage, ASAP performs coarse-to-fine region contraction within azimuthal strips to lock azimuth angles while retaining multiple maxima through spherical caps. In the second stage, it refines elevation along the great-circle arc between two close candidates. Extensive simulations and real-world experiments validate the efficiency and merits of the proposed method over existing approaches.

Low-rank matrix optimization is often carried out via the Burer-Monteiro (BM) formulation, but choosing the factorization rank $r$ is delicate and can substantially slow optimization. We propose a unified framework, termed direction-magnitude decomposition (DMD), that decomposes the optimization variable to improve optimization efficiency even when the target rank is unknown. We develop two DMD-based approaches and establish their theoretical advantages on the canonical problem of matrix factorization. The first, overparameterized DMD, uses a rank $r$ larger than necessary and enjoys faster convergence as $r$ increases. The second, recursive DMD, is motivated by the incremental eigenpair learning, or saddle-to-saddle, behavior of overparameterized DMD. It achieves lower memory and computational costs, complementing overparameterized DMD. Both approaches are exponentially faster than gradient descent applied to the BM formulation. Numerical experiments on matrix factorization, sensing, and completion corroborate our theoretical findings and demonstrate the practical effectiveness of DMD.