Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Achieving Optimal Sample Complexity for a Broader Class of Signals in Sparse Phase Retrieval

Mengchu Xu, Yuxuan Zhang, Jian Wang

Abstract

Sparse phase retrieval aims to recover a $k$-sparse signal from $m$ phaseless measurements. While the theoretically optimal sample complexity for successful recovery is $Ω(k \log n)$, existing algorithms can only achieve this bound for signals with specific structural assumptions, leading to a notable gap between theory and practice. To bridge this gap, we introduce an efficient initialization algorithm, termed generalized Exponential Spectral Pursuit (gESP). We prove that gESP can significantly expand the family of signals that are guaranteed to be recovered with the optimal sample complexity, thereby extending the scope of theoretical optimality to a much broader class of signals. Extensive simulations validate our theoretical findings and demonstrate that gESP consistently outperforms the state-of-the-art methods across diverse signal types.

Achieving Optimal Sample Complexity for a Broader Class of Signals in Sparse Phase Retrieval

Abstract

Sparse phase retrieval aims to recover a -sparse signal from phaseless measurements. While the theoretically optimal sample complexity for successful recovery is , existing algorithms can only achieve this bound for signals with specific structural assumptions, leading to a notable gap between theory and practice. To bridge this gap, we introduce an efficient initialization algorithm, termed generalized Exponential Spectral Pursuit (gESP). We prove that gESP can significantly expand the family of signals that are guaranteed to be recovered with the optimal sample complexity, thereby extending the scope of theoretical optimality to a much broader class of signals. Extensive simulations validate our theoretical findings and demonstrate that gESP consistently outperforms the state-of-the-art methods across diverse signal types.

Paper Structure

This paper contains 24 sections, 12 theorems, 120 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Algorithms
  3. Prior work
  4. Generalized Exponential Spectral Pursuit
  5. Theoretical Analysis
  6. Preliminary
  7. Theoretical results for sample complexity of gESP
  8. Discussion
  9. Broader Conditions for Optimal Sample Complexity
  10. Strict Superiority of Corollary \ref{['coro:3']} over Corollary \ref{['coro:2']}
  11. Proof
  12. Numerical Simulations
  13. Experimental Setup
  14. Performance Comparison
  15. Robustness to the Choice of Parameter $p$
  16. ...and 9 more sections

Key Result

Theorem 1

For any constant $0 < \delta < 1$, the output $\mathbf{z}$ of Algorithm alg:gESP satisfies $\mathop{\mathrm{dist}}\limits(\mathbf{z},\mathbf{x}) \leq \delta \|\mathbf{x}\|$ with probability at least $1 - n^{-c}$. The corresponding required sample complexity depends on an input parameter $p \in [k]$; To achieve the most efficient sampling, this complexity can be minimized by selecting the optimal p

Figures (4)

  • Figure 1: An illustration of the relaxed condition for achieving the optimal sample complexity $\Omega(k\log n)$. While state-of-the-art methods require the signal's energy to be concentrated in its single largest entry: $s(1) = \Theta(1)$, gESP achieves optimality for a broader class of signals where the energy is concentrated among the top $\lceil \sqrt{k} \rceil$ entries: $s(\lceil \sqrt{k} \rceil)= \Theta(1)$.
  • Figure 2: An Illustration of the two functions in \ref{['eq:gESP_k_result_2']}.
  • Figure 3: Performance comparison of relative error and fraction of recovered support as a function of sample ratio.
  • Figure 4: Performance comparison of gESP with different choices of $p$ under the designed power-law signal. We set $n=1000, k=100$ with $m/n$ varying from $0.05$ to $1.0$. The multi-peak strategies ($p > 1$) significantly outperform the single-peak baseline. Notably, the algorithm is robust to large $p$, with the full support choice ($p=k$) achieving the best empirical performance.

Theorems & Definitions (31)

  • Definition 1
  • Definition 2
  • Theorem 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Corollary 1
  • Remark 4
  • Corollary 2
  • Corollary 3
  • ...and 21 more