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

TL;DR

This paper introduces CPGD, a novel convergent gradient descent method for generalized FRI signal reconstruction, leveraging Cadzow denoising to efficiently handle non-convex optimization with improved stability and performance.

Contribution

It proposes a new implicit formulation of genFRI, and develops CPGD, an inexact proximal gradient method using Cadzow denoising, with proven convergence guarantees.

Findings

CPGD outperforms existing methods in non-uniform sampling scenarios.

The method demonstrates stable and accurate signal recovery.

Extensive simulations confirm the effectiveness of CPGD.

Abstract

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.