We develop a multidimensional version of Gradient-MUSIC for estimating the frequencies of a nonharmonic signal from noisy samples. The guiding principle is that frequency recovery should be based only on the signal subspace determined by the data. From this viewpoint, the MUSIC functional is an economical nonconvex objective encoding the relevant information, and the problem becomes one of understanding the geometry of its perturbed landscape. Our main contribution is a general structural theory showing that, under explicit conditions on the measurement kernel and the perturbation of the signal subspace, the perturbed MUSIC function is an admissible optimization landscape: suitable initial points can be found efficiently by coarse thresholding, gradient descent converges to the relevant local minima, and these minima obey quantitative error bounds. Thus the theory is not merely existential; it provides a constructive global optimization framework for multidimensional optimal resolution. We verify the abstract conditions in detail for two canonical sampling geometries: discrete samples on a cube and continuous samples on a ball. In both cases we obtain uniform, nonasymptotic recovery guarantees under deterministic as well as stochastic noise. In particular, for lattice samples in a cube of side length $4m$, if the true frequencies are separated by at least $β_d/m$ and the noise has $\ell^\infty$ norm at most $\varepsilon$, then Gradient-MUSIC recovers the frequencies with error at most \[ C_d \frac{\varepsilon}{m}, \] where $C_d, β_d>0$ depend only on the dimension. This scaling is minimax optimal in $m$ and $\varepsilon$. Under stationary Gaussian noise, the error improves to \[ C_d\frac{σ\sqrt{\log(m)}}{m^{1+d/2}}. \] This is the noisy super-resolution scaling: (see paper for rest of abstract)

翻译：我们提出了Gradient-MUSIC算法的多维版本，用于从含噪样本中估计非谐波信号的频率。其指导原则是：频率恢复应仅基于数据确定的信号子空间。基于这一视角，MUSIC泛函是一个经济的非凸目标函数，编码了相关信息，而问题转化为理解其扰动景观的几何结构。我们的主要贡献在于建立了一个通用结构理论：在测量核与信号子空间扰动的显式条件下，扰动后的MUSIC函数构成可容许的优化景观——可通过粗阈值化高效寻找合适的初始点，梯度下降收敛至相关局部极小点，且这些极小点满足定量误差界。因此该理论并非仅存在性结论：它为多维最优分辨率提供了构造性全局优化框架。我们针对两种经典采样几何（立方体上的离散样本和球体上的连续样本）详细验证了抽象条件。在两种情形下，我们均在确定性与随机噪声下获得了均匀的非渐近恢复保证。特别地，对于边长为$4m$的立方体中的格点样本，若真实频率间隔至少为$β_d/m$且噪声$\ell^\infty$范数不超过$\varepsilon$，则Gradient-MUSIC算法恢复频率的误差至多为\[ C_d \frac{\varepsilon}{m} \]其中$C_d,β_d>0$仅依赖于维度。该标度在$m$和$\varepsilon$上达到极小极大最优。在平稳高斯噪声下，误差改善为\[ C_d\frac{σ\sqrt{\log(m)}}{m^{1+d/2}} \]这对应于含噪超分辨率标度：（见原文剩余摘要）