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

Jinchi Chen

TL;DR

This work tackles simultaneous blind super-resolution and demixing of multiple point-source signals from low-frequency observations with unknown PSFs. It reformulates the problem as a block-diagonal low-rank matrix demixing task using a weighted vectorized Hankel lift and develops Scaled-GD, a scalable non-convex algorithm that forgoes explicit norm-balancing regularization. The authors prove linear convergence from spectral initialization with a sample complexity of $n \ge C_{\gamma} K^{2}s^{2}r^{2}\kappa^{2}\mu_{0}\mu_{1}\log^{2}(sn)$, independent of the condition number $\kappa$, and demonstrate competitive recovery performance against convex methods like ANM and VHL while achieving higher computational efficiency. Numerical experiments corroborate the theoretical guarantees, show robustness to noise, and validate practical recovery of both signal locations and PSFs, including a MUSIC-based localization step when desired.

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.

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

TL;DR

This work tackles simultaneous blind super-resolution and demixing of multiple point-source signals from low-frequency observations with unknown PSFs. It reformulates the problem as a block-diagonal low-rank matrix demixing task using a weighted vectorized Hankel lift and develops Scaled-GD, a scalable non-convex algorithm that forgoes explicit norm-balancing regularization. The authors prove linear convergence from spectral initialization with a sample complexity of , independent of the condition number , and demonstrate competitive recovery performance against convex methods like ANM and VHL while achieving higher computational efficiency. Numerical experiments corroborate the theoretical guarantees, show robustness to noise, and validate practical recovery of both signal locations and PSFs, including a MUSIC-based localization step when desired.

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.
Paper Structure (19 sections, 19 theorems, 120 equations, 5 figures, 1 algorithm)

This paper contains 19 sections, 19 theorems, 120 equations, 5 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Related Work and Contributions
  4. Organization and Notation
  5. Algorithm
  6. Main Results
  7. Numerical experiments
  8. Proof of Main Results
  9. Proof of Theorem \ref{['theorem: iteration']}
  10. Conclusion
  11. Proof of key lemmas
  12. Proof of Lemma \ref{['smooth of g']}
  13. Proof of Lemma \ref{['bound of nabla g']}
  14. Proof of Lemma \ref{['upper bound of h']}
  15. Proof of Lemma \ref{['lemma bound L']}
  16. ...and 4 more sections

Key Result

Theorem 3.1

Suppose that Assumption assumption 0 and Assumption assumption 1 hold. Let $\eta_t\leq\frac{1}{20}$. If the number of measurements satisfies that $n\geq C_{\gamma} K^2s^2r^2 \kappa^2\mu_0 \mu_1 \log^2(sn)$, then with probability at least $1-(sn)^{-\gamma}$, the sequence $\{\bm{L}_{k,t}, \bm{R}_{k,t} where $\sigma_0= \sqrt{\sum_{k=1}^{K}\sigma_r^2(\bm{Z}_{k, \natural}) }$ and $\kappa =\frac{\max_k

Figures (5)

  • Figure 1: The phase transitions of VHL, ANM and Scaled-GD were examined under the conditions $n=48$ and $K=2$. Top: the point source signal locations are randomly generated. Bottom: the locations obey the separation condition $\Delta:= \min_{k\neq j}\left|\tau_k-\tau_j\right|\geq\frac{1}{n}$. The red curve in these figures represents the hyperbola curve $rs=6$.
  • Figure 2: The phase transition of Scaled-GD for varying $n$ and $K$ when $s=r=3$. The red line in the figure represents the straight line $n=cK$, where $c$ is a constant.
  • Figure 3: The relative recovery errors of Scaled-GD and GD with respect to the iterations associated with different condition number $\kappa=1,5,15,20$. Left: $n=128, s=r=2, K=4$. Right: $n=512, s=r=4, K=6$.
  • Figure 4: Performance of Scaled-GD under different SNR.
  • Figure 5: Source localization and point spread function recovery with Scaled-GD and smoothed MUSIC. (a): Locations of point source signals and their estimates via smoothed MUSIC when $n=256, s=r=2, K=4$. (b) The magnitudes of $\{\bm{g}_{k,p}\}_{k=1,2; p=1,\cdots, 4}$ and their estimates obtained through least squares.

Theorems & Definitions (31)

  • Remark 3.1
  • Remark 3.2
  • Theorem 3.1
  • Remark 3.3
  • Remark 3.4
  • Remark 3.5
  • Lemma 5.1: Lipschitz smoothness of $g$
  • Lemma 5.2: Polyak-Lojasiewicz like inequality of $g$
  • Lemma 5.3
  • Lemma 5.4
  • ...and 21 more