AI 中文总结
研究多快照尖峰反卷积问题，采用变量投影公式将其转化为非凸最小二乘问题，刻画了目标函数凸性区域，证明估计器在不同噪声下的性质及梯度下降收敛性，数值实验验证了理论结果。
AI 中文摘要
我们研究多快照尖峰反卷积问题，目标是从已知点扩散函数（PSF）在多个快照上的带噪卷积中恢复稀疏脉冲的位置。我们采用变量投影公式，以封闭形式消除幅度，将任务简化为仅关于尖峰位置的非凸最小二乘问题，即尖峰反卷积的变量投影公式（VarProSD）。我们根据关键的PSF属性，包括其功率谱密度和平滑度，对VarProSD目标的凸性区域进行了明确刻画，揭示了采样带宽和尖峰分离如何影响局部几何结构。在这个区域内，我们证明了估计器在随机噪声下对于快照数量是一致的，并通过逆映射的局部Lipschitz属性在对抗噪声下提供了更精确的误差界。我们还展示了当初始化在该区域内时梯度下降的局部收敛保证。贯穿始终的一个核心要素是使用Beurling - Selberg极值逼近，它能对优化场景中出现的结构化矩阵的条件数给出与PSF无关的精确界。数值实验验证了我们的理论发现，并证明了改进的ESPRIT初始化随后基于梯度细化的有效性。
英文摘要
The problem of multi-snapshot spike deconvolution is studied, where the goal is to recover the locations of sparse impulses from their noisy convolution with a known point spread function (PSF) across multiple snapshots. A variable-projection formulation is adopted, in which the amplitudes are eliminated in closed form, thereby reducing the task to a nonconvex least-squares problem over the spike locations alone. This formulation is referred to as the variable-projection formulation of spike deconvolution (VarProSD). An explicit characterization of the basin of convexity of the VarProSD objective is provided in terms of key PSF properties, including its power spectral density and smoothness, revealing how sampling bandwidth and spike separation affect the local geometry. Within this basin, consistency of the estimator in the number of snapshots is established under stochastic noise, and a complementary, sharper error bound is derived under adversarial noise through the local Lipschitz property of the inverse map. Local convergence guarantees for gradient descent are further established when initialization is performed within the basin. A central role throughout the analysis is played by Beurling--Selberg extremal approximations, which enable sharp, PSF-agnostic bounds on the conditioning of the structured matrices arising in the optimization landscape. Numerical experiments are presented to corroborate the theoretical findings and demonstrate the effectiveness of modified ESPRIT initialization followed by gradient-based refinement.
