基于预条件一阶方法的ESPRIT在尖峰反卷积中的全局收敛性
Global Convergence of ESPRIT with Preconditioned First-Order Methods for Spike Deconvolution
AI总结：
针对尖峰反卷积问题，本文提出结合ESPRIT与自适应预条件梯度下降的新方法，通过非线性最小二乘估计点源参数，证明其超线性收敛速率及一维多快照下的全局收敛性，数值模拟验证了方法的鲁棒性与有效性。
摘要：
尖峰反卷积是从已知点扩散函数的卷积结果中恢复点源的问题，在许多传感和成像应用中起着基础性作用。本文提出一种结合ESPRIT与预条件梯度下降（PGD）的新方法，通过非线性最小二乘法估计点源的振幅和位置。预条件矩阵被自适应设计以适应学习过程中的变化，确保已证明的超线性收敛速率。我们提供PGD的局部收敛保证和ESPRIT重建的性能分析，进而在一维多快照设置下为我们的方法提供全局收敛保证，证明其鲁棒性和有效性。数值模拟证实了所提尖峰反卷积方法的性能。
英文摘要：
Spike deconvolution is the problem of recovering point sources from their convolution with a known point spread function, playing a fundamental role in many sensing and imaging applications. This paper proposes a novel approach combining ESPRIT with Preconditioned Gradient Descent (PGD) to estimate the amplitudes and locations of the point sources by a non-linear least squares. The preconditioning matrices are adaptively designed to account for variations in the learning process, ensuring a proven super-linear convergence rate. We provide local convergence guarantees for PGD and performance analysis of ESPRIT reconstruction, leading to global convergence guarantees for our method in one-dimensional settings with multiple snapshots, demonstrating its robustness and effectiveness. Numerical simulations corroborate the performance of the proposed approach for spike deconvolution.
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