Blind Super-resolution of Point Sources via Projected Gradient Descent

TL;DR

This paper introduces a non-convex projected gradient descent method for blind super-resolution of point sources, leveraging low rank matrix recovery and Hankel matrix structures, with proven convergence and competitive performance.

Contribution

It proposes a simple, efficient non-convex algorithm for blind super-resolution based on low rank Hankel matrix recovery, with theoretical convergence guarantees.

Findings

Method converges linearly to the target matrix.

Achieves comparable recovery performance to convex methods.

Demonstrates efficiency and effectiveness through numerical experiments.

Abstract

Blind super-resolution can be cast as a low rank matrix recovery problem by exploiting the inherent simplicity of the signal and the low dimensional structure of point spread functions. In this paper, we develop a simple yet efficient non-convex projected gradient descent method for this problem based on the low rank structure of the vectorized Hankel matrix associated with the target matrix. Theoretical analysis indicates that the proposed method exactly converges to the target matrix with a linear convergence rate under the similar conditions as convex approaches. Numerical results show that our approach is competitive with existing convex approaches in terms of recovery ability and efficiency.