Spike deconvolution is the problem of recovering the point sources from their convolution with a known point spread function, which plays a fundamental role in many sensing and imaging applications. In this paper, we investigate the local geometry of recovering the parameters of point sources$\unicode{x2014}$including both amplitudes and locations$\unicode{x2014}$by minimizing a natural nonconvex least-squares loss function measuring the observation residuals. We propose preconditioned variants of gradient descent (GD), where the search direction is scaled via some carefully designed preconditioning matrices. We begin with a simple fixed preconditioner design, which adjusts the learning rates of the locations at a different scale from those of the amplitudes, and show it achieves a linear rate of convergence$\unicode{x2014}$in terms of entrywise errors$\unicode{x2014}$when initialized close to the ground truth, as long as the separation between the true spikes is sufficiently large. However, the convergence rate slows down significantly when the dynamic range of the source amplitudes is large. To bridge this issue, we introduce an adaptive preconditioner design, which compensates for the learning rates of different sources in an iteration-varying manner based on the current estimate. The adaptive design provably leads to an accelerated convergence rate that is independent of the dynamic range, highlighting the benefit of adaptive preconditioning in nonconvex spike deconvolution. Numerical experiments are provided to corroborate the theoretical findings.

翻译：尖峰反卷积是指从已知点扩散函数与点源的卷积中恢复点源的问题，这在许多传感与成像应用中具有基础性作用。本文研究通过最小化度观测残差的自然非凸最小二乘损失函数，恢复点源参数（包括振幅和位置）的局部几何性质。我们提出梯度下降（GD）的预条件变体，其中搜索方向通过精心设计的预条件矩阵进行缩放。首先采用简单的固定预条件设计，以不同尺度调整位置与振幅的学习率，并证明当初始化接近真实值且真实尖峰之间的间隔足够大时，该方法在逐元素误差上可实现线性收敛率。然而，当源振幅的动态范围较大时，收敛速度会显著减慢。为解决此问题，我们引入自适应预条件设计，该设计基于当前估计值以迭代变化的方式补偿不同源的学习率。该自适应设计可证明实现与动态范围无关的加速收敛率，突显了自适应预条件在非凸尖峰反卷积中的优势。数值实验验证了理论结果。