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

This is the code for "Simpler Gradient Methods for Blind Super-Resolution with Lower Iteration Complexity" by Jinsheng Li, Wei Cui, Xu Zhang, in IEEE Transactions on Signal Processing, DOI: 10.1109/TSP.2024.3470071, arxiv.

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.

Algorithms

Experiments

Some experiments in our paper.

Error versus iteration

Error versus runtime

Recovery in a 2D real case

Citation

If you find this code useful for your research, please consider citing:

@ARTICLE{10697932,
  author={Li, Jinsheng and Cui, Wei and Zhang, Xu},
  journal={IEEE Transactions on Signal Processing}, 
  title={Simpler Gradient Methods for Blind Super-Resolution with Lower Iteration Complexity}, 
  year={2024},
  volume={},
  number={},
  pages={1-16},
  keywords={Superresolution;Complexity theory;Convergence;Minimization;Gradient methods;Signal processing algorithms;Sensors;Optimization;Vectors;Radar imaging;Blind super-resolution;vanilla gradient descent;scaled gradient descent;low-rank matrix factorization},
  doi={10.1109/TSP.2024.3470071}}

Code descriptions

main, 1D case

demo.m
demo how to run our simpler gradient methods VGD-VHL, ScalGD-VHL , in 1-D signal case.

solverPgd_fh.m
PGD-VHL algorithm, 1D case, fast computing via Hankel structure

solverVgd_fh.m
VGD-VHL algorithm, 1D case, fast computing via Hankel structure

solverScaledgd_fh.m
ScalGD-VHL algorithm, 1D case, fast computing via Hankel structure

generateSignals_bdft_withsep.m
generate the true signal and measurements.

./2Dcase

demo2d.m

run a fixed 2D case via our algorithms, and plot the delay-doppler location via 2D MUSIC

real2Dcase.mat
true signal matrix; a fixed case.

solverPgd2d.m
PGD-VHL algorithm, 2D case

solverVgd2d.m
VGD-VHL algorithm, 2D case

solverScaledgd2d.m
ScalGD-VHL algorithm, 2D case

music_2d.m
MUSIC for 2D signal super-resolution

getSignals_ofdm.m
generate 2-D signal