CPGD：Cadzow即插即用梯度下降，适用于通用FRI
CPGD: Cadzow Plug-and-Play Gradient Descent for Generalised FRI
摘要：
有限创新率（FRI）是一个强大的重建框架，使稀疏狄拉克流从均匀的低通滤波样本的恢复。这个框架的扩展，被称为广义FRI（genFRI），最近已被提出用于处理任意线性测量模型的情况。在这种情况下，信号重建相当于解决一个联合约束优化问题，产生狄拉克流的傅里叶级数系数和它的所谓的零化滤波器的估计，涉及正则化项。然而，这个优化问题在数据中是高度非凸和非线性的。此外，建议的数值求解器是计算密集型的，没有收敛保证。在这项工作中，我们提出了一个隐式制定的genFRI问题。为此，我们利用一种新的正则化项，它不显式地依赖于未知的零化滤波器，但在稳定恢复的解决方案中强制执行足够的结构。由此产生的优化问题仍然是非凸的，但更简单，因为线性的数据和更少的未知数。我们通过可证明收敛的近似梯度下降（PGD）方法来解决它。由于近端步骤不承认一个简单的封闭形式的表达，我们提出了一个不精确的PGD方法，创造Cadzow即插即用梯度下降（CPGD）。后者通过Cadzow去噪（FRI中的一种众所周知的去噪算法）来近似最接近的步骤。我们为CPGD提供了局部不动点收敛保证。通过大量的数值模拟，我们证明了在非均匀时间样本的情况下，CPGD相对于现有技术的优越性。
英文摘要：
Finite rate of innovation (FRI) is a powerful reconstruction framework enabling the recovery of sparse Dirac streams from uniform low-pass filtered samples. An extension of this framework, called generalised FRI (genFRI), has been recently proposed for handling cases with arbitrary linear measurement models. In this context, signal reconstruction amounts to solving a joint constrained optimisation problem, yielding estimates of both the Fourier series coefficients of the Dirac stream and its so-called annihilating filter, involved in the regularisation term. This optimisation problem is however highly non convex and non linear in the data. Moreover, the proposed numerical solver is computationally intensive and without convergence guarantee. In this work, we propose an implicit formulation of the genFRI problem. To this end, we leverage a novel regularisation term which does not depend explicitly on the unknown annihilating filter yet enforces sufficient structure in the solution for stable recovery. The resulting optimisation problem is still non convex, but simpler since linear in the data and with less unknowns. We solve it by means of a provably convergent proximal gradient descent (PGD) method. Since the proximal step does not admit a simple closed-form expression, we propose an inexact PGD method, coined as Cadzow plug-and-play gradient descent (CPGD). The latter approximates the proximal steps by means of Cadzow denoising, a well-known denoising algorithm in FRI. We provide local fixed-point convergence guarantees for CPGD. Through extensive numerical simulations, we demonstrate the superiority of CPGD against the state-of-the-art in the case of non uniform time samples.
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