We introduce the Gradient-MUSIC algorithm for estimating the unknown frequencies and amplitudes of a nonharmonic signal from noisy time samples. While the classical MUSIC algorithm performs a computationally expensive search over a fine grid, Gradient-MUSIC is significantly more efficient and eliminates the need for discretization over a fine grid by using optimization techniques. It coarsely scans the 1D landscape to find initialization simultaneously for all frequencies followed by parallelizable local refinement via gradient descent. We also analyze its performance when the noise level is sufficiently small and the signal frequencies are separated by at least $8\pi/m$, where $\pi/m$ is the standard resolution of this problem. Even though the 1D landscape is nonconvex, we prove a global convergence result for Gradient-MUSIC: coarse scanning provably finds suitable initialization and gradient descent converges at a linear rate. In addition to convergence results, we also upper bound the error between the true signal frequencies and amplitudes with those found by Gradient-MUSIC. For example, if the noise has $\ell^\infty$ norm at most $\varepsilon$, then the frequencies and amplitudes are recovered up to error at most $C\varepsilon/m$ and $C\varepsilon$ respectively, which are minimax optimal in $m$ and $\varepsilon$. Our theory can also handle stochastic noise with performance guarantees under nonstationary independent Gaussian noise. Our main approach is a comprehensive geometric analysis of the landscape, a perspective that has not been explored before.

翻译：本文提出Gradient-MUSIC算法，用于从含噪声的时间采样数据中估计非谐波信号的未知频率与振幅。经典MUSIC算法需在精细网格上进行计算代价高昂的搜索，而Gradient-MUSIC通过优化技术显著提升效率，无需依赖精细网格离散化。该算法首先对一维参数空间进行粗粒度扫描，为所有频率参数同步寻找初始化值，随后通过可并行化的梯度下降法进行局部精细化。我们分析了该算法在噪声水平足够小且信号频率间隔至少为$8\pi/m$（其中$\pi/m$为本问题的标准分辨率）时的性能。尽管一维参数空间具有非凸性，我们证明了Gradient-MUSIC具有全局收敛性：粗粒度扫描可确保找到合适的初始化值，且梯度下降以线性速率收敛。除收敛性证明外，我们还建立了真实信号频率/振幅与Gradient-MUSIC估计值之间误差的上界。例如，当噪声的$\ell^\infty$范数至多为$\varepsilon$时，频率与振幅的恢复误差分别不超过$C\varepsilon/m$与$C\varepsilon$，该误差界在$m$和$\varepsilon$意义上达到极小极大最优。我们的理论框架同样适用于随机噪声场景，在非平稳独立高斯噪声下仍能保证性能。本研究的核心方法是对参数空间几何结构的系统性分析，这一研究视角在以往工作中尚未被深入探索。