Fast and Provable Simultaneous Blind Super-Resolution and Demixing for Point Source Signals: Scaled Gradient Descent without Regularization

TL;DR

This paper introduces a fast, provably convergent scaled gradient descent method for simultaneous blind super-resolution and demixing of point source signals from limited spectral data, without regularization.

Contribution

It develops a novel regularization-free scaled gradient descent algorithm with theoretical guarantees for blind super-resolution and demixing, improving efficiency and accuracy.

Findings

Linear convergence to ground truth with spectral initialization

Competitive recovery accuracy compared to convex methods

Efficient algorithm without regularization

Abstract

We address the problem of simultaneously recovering a sequence of point source signals from observations limited to the low-frequency end of the spectrum of their summed convolution, where the point spread functions (PSFs) are unknown. By exploiting the low-dimensional structures of the signals and PSFs, we formulate this as a low-rank matrix demixing problem. To solve this, we develop a scaled gradient descent method without balancing regularization. We establish theoretical guarantees under mild conditions, demonstrating that our method, with spectral initialization, converges to the ground truth at a linear rate, independent of the condition number of the underlying data matrices. Numerical experiments indicate that our approach is competitive with existing convex methods in terms of both recovery accuracy and computational efficiency.