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

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 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.

TV-based Reconstruction of Periodic Functions

“…where x ∈ Z is the largest integer smaller or equal to x. To show this, it suffices to apply (D + αId) on both sides of (53). Similar formulas can be obtained for any N ≥ 1.…”

“…Optimization in periodic function spaces. Several works for Dirac recovery have been developed over the torus, and are therefore tailored for periodic Dirac streams [18,20,23,24,26,32,53]. Contrarily to the non periodic setting, the extension to arbitrary periodic functions has only received a limited attention so far, mostly in the works of Simeoni [43,44].…”

“…As a potential cure to this issue, one could consider meshfree algorithms capable of directly recovering the non uniform knots in the continuum. Candidate reconstruction algorithms include the Cadzow Plug-and-Play Gradient Descent (CPGD) algorithm [53] as well as the Franck-Wolfe algorithm and its variants [33,36]. Both algorithms have been successfully used for the reconstruction of periodic Dirac streams.…”

Sentiment Analysis of the Consumer Review Text Based on BERT-BiLSTM in a Social Media Environment

“…The objective is to find the parameters that minimize the empirical risk. To do so, the Fast Gradient Method (FGM) (Florea & Vorobyov, 2020) and the Projection Gradient Method (PGD) (Simeoni et al, 2021) are selected as the adversarial training methods:…”

Time Encoding of Finite-Rate-of-Innovation Signals

“…Higher the σ, more is the jitter. We consider reconstruction using GenFRI-TEM (Algorithm 1), FRI-TEM [24] and Cadzow Plug-and-Play Gradient Descent (CPGD) [36] techniques. The…”