Simpler Gradient Methods for Blind Super-Resolution with Lower Iteration Complexity

TL;DR

This paper introduces simpler gradient methods for blind super-resolution that reduce iteration complexity and improve computational efficiency without sacrificing recovery accuracy.

Contribution

Proposes two new gradient algorithms, VGD-VHL and ScalGD-VHL, with lower iteration complexity and less reliance on regularization compared to previous methods.

Findings

Lower iteration complexity than PGD-VHL

ScalGD-VHL has the lowest iteration complexity, independent of condition number

Methods achieve comparable recovery performance with improved efficiency

Abstract

We study the problem of blind super-resolution, which can be formulated as a low-rank matrix recovery problem via vectorized Hankel lift (VHL). The previous gradient descent method based on VHL named PGD-VHL relies on additional regularization such as the projection and balancing penalty, exhibiting a suboptimal iteration complexity. In this paper, we propose a simpler unconstrained optimization problem without the above two types of regularization and develop two new and provable gradient methods named VGD-VHL and ScalGD-VHL. A novel and sharp analysis is provided for the theoretical guarantees of our algorithms, which demonstrates that our methods offer lower iteration complexity than PGD-VHL. In addition, ScalGD-VHL has the lowest iteration complexity while being independent of the condition number. Furthermore, our novel analysis reveals that the blind super-resolution problem is less incoherence-demanding, thereby eliminating the necessity for incoherent projections to achieve linear convergence. Empirical results illustrate that our methods exhibit superior computational efficiency while achieving comparable recovery performance to prior arts.