发表机构
School of Mathematical Sciences, Zhejiang University; Institute of Fundamental and Transdisciplinary Research, Zhejiang University（浙江大学数学学院; 浙江大学基础与交叉科学研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出用于阵列信号处理的广义汉克尔/托普利茨矩阵框架，推导了高维超分辨率源数检测的计算分辨率极限上界，解决了传统方法的高维计算瓶颈，提升了非均匀测量下的源定位性能。
AI 中文摘要
本文针对非均匀阵列信号处理和多维超分辨率问题，引入了广义汉克尔/托普利茨矩阵（GHM/GTM）及相关的广义范德蒙德分解。该框架源于分辨率极限理论的研究，在保留底层低秩范德蒙德分解的同时，突破了经典汉克尔/托普利茨结构的限制，允许采用更灵活的采样几何。基于GHM框架设计的最优算法，我们推导了通用d维超分辨率问题中源数检测的计算分辨率极限（CRL）的当前最优上界估计。对于与稀疏分布式阵列几何密切相关的分段采样集，我们建立了关联广义范德蒙德矩阵最小奇异值的确定性下界，并推导了多簇源配置对应的稳定性和源数检测保证。为解决高维场景下传统多级汉克尔构造的计算瓶颈，我们进一步引入随机GHM构造，其矩阵维度随有效自由度而非完整张量积网格缩放，并在已实现的范德蒙德因子条件下给出确定性恢复保证。我们还将该框架扩展至源定位，开发了适用于非均匀测量的GHM基MUSIC算法，其稳定性通过广义范德蒙德因子的条件数表征。合成数据的数值实验表明，所提GHM基方法在达到有竞争力的分辨率和恢复精度的同时，大幅减小了矩阵规模并降低了计算成本，尤其在高维场景下表现突出。
英文摘要
In this paper, we introduce generalized Hankel/Toeplitz matrices (GHM/GTM) and the associated generalized Vandermonde decomposition for nonuniform array signal processing and multi-dimensional super-resolution. The proposed framework was discovered from the study of resolution limit theory and extends the classical Hankel/Toeplitz structure by allowing substantially more flexible sampling geometries while preserving the underlying low-rank Vandermonde factorization. Through devising an optimal algorithm based on this GHM framework, we derive the state-of-the-art upper bound estimate for the computational resolution limit (CRL) of source-number detection in general $d$-dimensional super-resolution problems. For segmented sampling sets, whose geometry is closely related to sparse and distributed arrays, we establish deterministic lower bounds for the minimum singular values of the associated generalized Vandermonde matrices and derive corresponding stability and number-detection guarantees for multi-clump source configurations. To address the computational bottleneck of conventional multi-level Hankel constructions in high dimensions, we further introduce randomized GHM constructions whose matrix dimensions scale with the effective degrees of freedom rather than with the full tensor-product grid, together with deterministic recovery guarantees conditional on the realized Vandermonde factors. We also extend the framework to source localization by developing GHM-based MUSIC algorithms for nonuniform measurements, with stability characterized through the conditioning of the generalized Vandermonde factors. Numerical experiments on synthetic data demonstrate that the proposed GHM-based methods achieve competitive resolution and recovery accuracy while substantially reducing matrix size and computational cost, especially in high-dimensional settings.
Comments48 pages, 13 figures
