用于广义创新有限率的Cadzow投影梯度下降：定量局部收敛理论

中文摘要 AI 辅助

广义创新有限率（GenFRI）框架旨在重构通过含噪线性测量模型测得的创新有限率（FRI）信号。GenFRI近期被重新表述为结构化低秩优化问题，Cadzow投影梯度下降（CPGD）算法被用于求解该问题。尽管CPGD在实际应用中表现良好，但仅建立了定性的局部收敛保证。我们结合去噪正则化框架重新研究GenFRI，将其重新表述为解位于Cadzow去噪子不动点集的优化问题。我们证明该类算法中没有任何算法能获得全局保证，反而确定Cadzow去噪子在FRI模型集的显式邻域上是拟非扩张的，其半径由基础狄拉克流的条件数决定。基于这些结果，我们提出广义CPGD（GCPGD）算法，并证明其在显式吸引域内的任意初始化下均可收敛，且重构误差边界与噪声水平成比例。数值实验验证了预测的收缩率、恢复阈值和运行时间保证，结果显示单次运行GCPGD的性能优于当前最优的GenFRI算法。

英文摘要

The generalized finite rate of innovation~(GenFRI) framework aims at reconstructing finite-rate-of-innovation~(FRI) signals measured through a noisy linear measurement model. GenFRI has been recently recast as a structured low-rank optimization problem and the Cadzow projected gradient descent~(CPGD) algorithm has been suggested to solve it. While CPGD works well in practice, only qualitative local convergence guarantees have been established. We revisit GenFRI in the light of the regularization by denoising framework, recasting it as an optimization problem whose solutions lie in the fixed-point set of the Cadzow denoiser. We show that no algorithm in this family can enjoy global guarantees, and establish instead that the Cadzow denoiser is quasi-nonexpansive on an explicit neighborhood of the FRI model set, whose radius is governed by the conditioning of the underlying Dirac stream. Building on these results, we propose the generalized CPGD~(GCPGD) algorithm and prove its convergence from any initialization within an explicit basin of attraction, together with a reconstruction error bound proportional to the noise level. Numerical experiments validate the predicted contraction rates, recovery thresholds, and run-time certificates, and show that a single run of GCPGD outperforms state-of-the-art GenFRI algorithms.

发表机构

• E-Scopics SAS（E-Scopics 公司）

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