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Optimal sparse phase retrieval via a quasi-Bayesian approach

The Tien Mai

TL;DR

This work addresses sparse phase retrieval, where θ^* ∈ ℝ^p must be recovered from magnitude-only measurements. It introduces a sparse quasi-Bayesian framework using a Gibbs posterior with a continuous sparsity-promoting prior (scaled Student-t) and analyzes it via PAC-Bayesian bounds, achieving minimax-optimal rates under sub-exponential noise. The authors prove non-asymptotic risk bounds and posterior contraction, showing adaptivity to unknown sparsity s^* and matching known frequentist rates. For computation, gradient-based Langevin Monte Carlo sampling is employed, and numerical experiments demonstrate competitive performance with state-of-the-art frequentist methods, including applications to handwritten digit reconstruction, highlighting the method's practical viability as a principled Bayesian alternative for noisy sparse phase retrieval.

Abstract

This paper addresses the problem of sparse phase retrieval, a fundamental inverse problem in applied mathematics, physics, and engineering, where a signal need to be reconstructed using only the magnitude of its transformation while phase information remains inaccessible. Leveraging the inherent sparsity of many real-world signals, we introduce a novel sparse quasi-Bayesian approach and provide the first theoretical guarantees for such an approach. Specifically, we employ a scaled Student distribution as a continuous shrinkage prior to enforce sparsity and analyze the method using the PAC-Bayesian inequality framework. Our results establish that the proposed Bayesian estimator achieves minimax-optimal convergence rates under sub-exponential noise, matching those of state-of-the-art frequentist methods. To ensure computational feasibility, we develop an efficient Langevin Monte Carlo sampling algorithm. Through numerical experiments, we demonstrate that our method performs comparably to existing frequentist techniques, highlighting its potential as a principled alternative for sparse phase retrieval in noisy settings.

Optimal sparse phase retrieval via a quasi-Bayesian approach

TL;DR

This work addresses sparse phase retrieval, where θ^* ∈ ℝ^p must be recovered from magnitude-only measurements. It introduces a sparse quasi-Bayesian framework using a Gibbs posterior with a continuous sparsity-promoting prior (scaled Student-t) and analyzes it via PAC-Bayesian bounds, achieving minimax-optimal rates under sub-exponential noise. The authors prove non-asymptotic risk bounds and posterior contraction, showing adaptivity to unknown sparsity s^* and matching known frequentist rates. For computation, gradient-based Langevin Monte Carlo sampling is employed, and numerical experiments demonstrate competitive performance with state-of-the-art frequentist methods, including applications to handwritten digit reconstruction, highlighting the method's practical viability as a principled Bayesian alternative for noisy sparse phase retrieval.

Abstract

This paper addresses the problem of sparse phase retrieval, a fundamental inverse problem in applied mathematics, physics, and engineering, where a signal need to be reconstructed using only the magnitude of its transformation while phase information remains inaccessible. Leveraging the inherent sparsity of many real-world signals, we introduce a novel sparse quasi-Bayesian approach and provide the first theoretical guarantees for such an approach. Specifically, we employ a scaled Student distribution as a continuous shrinkage prior to enforce sparsity and analyze the method using the PAC-Bayesian inequality framework. Our results establish that the proposed Bayesian estimator achieves minimax-optimal convergence rates under sub-exponential noise, matching those of state-of-the-art frequentist methods. To ensure computational feasibility, we develop an efficient Langevin Monte Carlo sampling algorithm. Through numerical experiments, we demonstrate that our method performs comparably to existing frequentist techniques, highlighting its potential as a principled alternative for sparse phase retrieval in noisy settings.

Paper Structure

This paper contains 19 sections, 6 theorems, 60 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Problem and method
  3. Sparse phase retrieval
  4. A quasi-Bayesian approach
  5. Theoretical guarantees
  6. Assumptions
  7. Main results
  8. Numerical studies
  9. Implementation using a gradient-based sampling method
  10. Simulation setup
  11. Simulation results
  12. Sample size effect:
  13. Noise-to-signal effect:
  14. Sparsity effect:
  15. Tuning parameter effect:
  16. ...and 4 more sections

Key Result

Theorem 1

Assume that Assumption assume_X_bounded, assum_heavy_tailed and assume_mendelson are satisfied. Take $\lambda = \lambda^*$, $\varsigma = \varsigma^*$. Then for all $\theta^*$ such that $\| \theta^*\|_2 \leq H_1 - 2p \varsigma$, we have with probability at least $1-\delta, \delta\in (0,1)$ that for some universal constant $\mathfrak{C} > 0$ depending only on $H_1, C,\kappa_0, \xi$.

Figures (5)

  • Figure 1: The relation between the minimum relative error and the sample size.
  • Figure 2: The relation between the minimum relative error and the noise-to-signal-ratio.
  • Figure 3: The relation between the minimum relative error and the sparsity.
  • Figure 4: The relation between the minimum relative error and the tuning parameters. Left: changing $\varsigma$, Right: changing $\lambda$.
  • Figure 5: Handwritten Digit Images recovered by different methods.

Theorems & Definitions (11)

  • Theorem 1
  • Remark 1
  • Theorem 2
  • proof : Proof of Theorem \ref{['thm_heavy_tailed_loss']}
  • proof : Proof of Theorem \ref{['thrm_contraction_slow']}
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • proof
  • Lemma 4
  • ...and 1 more