论文标题

CPGD:CADZOW插件的梯度下降

CPGD: Cadzow Plug-and-Play Gradient Descent for Generalised FRI

论文作者

Simeoni, Matthieu, Besson, Adrien, Hurley, Paul, Vetterli, Martin

论文摘要

有限的创新速率(FRI)是一个强大的重建框架,可从均匀的低通滤波样品中恢复稀疏的狄拉克流。最近已经提出了该框架的扩展称为广义星期五(GenFri),用于处理任意线性测量模型的案例。在这种情况下,信号重建相当于解决关节约束优化问题,从而估算了狄拉克流的傅立叶串联系数及其所谓的歼灭过滤器,该滤网涉及正则化项。但是,该优化问题在数据中是高度非凸面和非线性的。此外,提出的数值求解器在计算上是密集的,没有收敛保证。 在这项工作中,我们提出了Genfri问题的隐式配方。为此,我们利用了一个新颖的正则化项,该术语不明确取决于未知的歼灭过滤器,但在解决方案中实施了足够的结构以稳定恢复。最终的优化问题仍然不是凸,但是由于数据中线性和未知数较低,因此更简单。我们通过可证明的收敛近端梯度下降(PGD)方法来解决它。由于近端步骤不接受简单的闭合形式表达式,因此我们提出了一种不精确的PGD方法,该方法被认为是Cadzow插件梯度下降(CPGD)。后者通过Cadzow Denoising近似近端步骤,这是一种著名的DeNoising算法。我们为CPGD提供本地的定点收敛保证。通过广泛的数值模拟,我们证明了在非统一时间样本的情况下,CPGD与最先进的优势。

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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