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

Jinsheng Li, Wei Cui, Xu Zhang

TL;DR

A novel analysis reveals that the blind super-resolution problem is less incoherence-demanding, thereby eliminating the necessity for incoherent projections to achieve linear convergence and two new and provable gradient methods named VGD-VHL and ScalGD-VHL are developed.

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.

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

TL;DR

A novel analysis reveals that the blind super-resolution problem is less incoherence-demanding, thereby eliminating the necessity for incoherent projections to achieve linear convergence and two new and provable gradient methods named VGD-VHL and ScalGD-VHL are developed.

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.
Paper Structure (35 sections, 18 theorems, 107 equations, 8 figures, 3 tables, 2 algorithms)

This paper contains 35 sections, 18 theorems, 107 equations, 8 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Motivation and Contributions
  3. Related work
  4. Problem formulation and Algorithms
  5. Problem formulation
  6. Algorithms
  7. Theoretical guarantees
  8. Preliminaries
  9. Challenges and main results
  10. Discussion
  11. Comparisons on the demand for incoherence
  12. Extension to other problems
  13. Numerical Simulations
  14. Phase transitions
  15. Convergence performance
  16. ...and 20 more sections

Key Result

Theorem 1

With probability at least $1-c_1(sn)^{-c_2}$, the iterate in Alg. alg:VGD satisfies provided $\eta\leq\frac{1}{25\sigma_1(\bm{M}_\star)}$ and $n\geq C_1\delta_0^{-2} \mu_0^2\mu_1s^2r^2\kappa^3\log^2(sn)$, where $\delta_0\leq\frac{2}{25}$, $C_1$ is a universal constant and $\kappa=\frac{\sigma_1(\bm{M}_\star)}{\sigma_r(\bm{M}_\star)}$.

Figures (8)

  • Figure 1: The convergence performance of PGD-VHL and vanilla gradient descent without projection and balancing, where $n$ is the length of the signal, $r$ is the number of point sources, and $s$ is the dimension of subspace the PSFs live in.
  • Figure 2: The comparisons of VHL, PGD-VHL, VGD-VHL, and ScalGD-VHL in terms of phase transitions. The red curve plots $sr=20$.
  • Figure 3: The relative errors of PGD-VHL, VGD-VHL, and ScalGD-VHL versus the iteration number.
  • Figure 4: The comparisons of VGD-VHL and ScalGD-VHL under different condition numbers of $\bm{M}_\star$ as $\kappa=1, 5, 15$.
  • Figure 5: The average running time required for PGD-VHL, VGD-VHL, and ScalGD-VHL towards different recovery accuracies.
  • ...and 3 more figures

Theorems & Definitions (36)

  • Definition 1: The scaled norm
  • Theorem 1: Exact recovery of VGD-VHL
  • Theorem 2: Exact recovery of ScalGD-VHL
  • Remark 1: Step-size strategy
  • Remark 2: Iteration complexity
  • Remark 3: Initialization
  • Theorem 3
  • Lemma 1: Approximate balancing
  • proof
  • Lemma 2: PL inequality
  • ...and 26 more