Optimal Rates of Convergence for Noisy Sparse Phase Retrieval via Thresholded Wirtinger Flow (1506.03382v1)

Abstract: This paper considers the noisy sparse phase retrieval problem: recovering a sparse signal $x \in \mathbb{R}^p$ from noisy quadratic measurements $y_j = (a_j' x )^2 + \epsilon_j$, $j=1, \ldots, m$, with independent sub-exponential noise $\epsilon_j$. The goals are to understand the effect of the sparsity of $x$ on the estimation precision and to construct a computationally feasible estimator to achieve the optimal rates. Inspired by the Wirtinger Flow [12] proposed for noiseless and non-sparse phase retrieval, a novel thresholded gradient descent algorithm is proposed and it is shown to adaptively achieve the minimax optimal rates of convergence over a wide range of sparsity levels when the $a_j$'s are independent standard Gaussian random vectors, provided that the sample size is sufficiently large compared to the sparsity of $x$.

• The paper introduces a thresholded Wirtinger Flow algorithm that combines spectral initialization with iterative thresholding to recover sparse signals.
• It establishes minimax optimal convergence rates with error bounds of order (σ/||x||₂)√(k log p/m) under noisy quadratic measurements.
• The method bridges computational efficiency and statistical optimality, offering actionable insights for high-dimensional sparse signal recovery.

Analysis of Optimal Rates of Convergence for Noisy Sparse Phase Retrieval via Thresholded Wirtinger Flow

The paper "Optimal Rates of Convergence for Noisy Sparse Phase Retrieval via Thresholded Wirtinger Flow" by Cai, Li, and Ma introduces a methodological and theoretical advancement in solving the noisy sparse phase retrieval problem, where the task is to recover a sparse signal given noisy quadratic measurements. This problem is pertinent in various applications such as X-ray crystallography and astronomy, where amplitude information is obtained but not the phase, making it a nonlinear and ill-posed problem.

Summary of Contributions and Methodology

The authors propose a novel approach based on the Wirtinger Flow methodology, extended to handle sparsity and noise. The core of their contribution is a thresholded gradient descent algorithm, termed the Thresholded Wirtinger Flow, that iteratively updates the estimated signal while enforcing sparsity through thresholding after each gradient step. This method is backed by a minimax optimal convergence rate analysis.

1. Problem Formulation: The noisy phase retrieval model is defined as recovering a sparse signal $\mathbf{x} \in \mathbb{R}^p$ from measurements $y_j = (\mathbf{a}_j' \mathbf{x})^2 + \epsilon_j$, where $\epsilon_j$ are independent sub-exponential noise terms, and the vectors $\mathbf{a}_j$ are drawn from a Gaussian distribution.
2. Thresholded Wirtinger Flow Algorithm: The proposed algorithm comprises two main steps:
   • Initialization: Utilizes a spectral method that feeds into a diagonal thresholding technique to obtain an accurate initial estimate.
   • Iterative Thresholding: Involves gradient descent followed by a threshold step to maintain sparsity.
3. Theoretical Guarantees: The algorithm is shown to achieve minimax optimal rates of convergence for signal recovery. Specifically, the results indicate that the estimation error under the $\ell_2$ norm is of order $$\frac{\sigma}{\|\mathbf{x}\|_2} \sqrt{\frac{k \log p}{m}}$$, where $k$ represents the sparsity and $\sigma$ the noise level.

Practical and Theoretical Implications

The contributions underscore the balance between computational tractability and statistical efficiency when addressing phase retrieval in noisy settings. By demonstrating the minimax optimality of the proposed algorithm, the authors ensure that the achievable rates are not merely theoretical bounds but can be realized in practice with their method.

The implications extend to a broader realization that non-convex methods, when coupled with iterative thresholding, can lead to significant improvements in computational efficiency and solution accuracy over traditional convex relaxation methods. This work highlights the potential for applying similar thresholding strategies to other structured estimation problems in high dimensions, pushing the envelope for feasible solutions in sparsity-constrained optimization.

Future Directions in AI and Sparse Recovery

Future research could investigate the adaptability of the thresholded Wirtinger Flow algorithm to other structured signal models, such as low-rank matrix recovery, by leveraging the interplay between structured priors and non-convex optimization. Additionally, the computational limits observed could be explored deeper by evaluating the performance of the proposed method against highly structured measurement models versus unstructured Gaussian designs.

The integration of prior information in refining the spectral initialization and thresholding criteria could also be examined to enhance robustness across various noise distributions and model mis-specifications. Furthermore, this methodology can inspire more advanced algorithms that combine deep learning approaches with traditional signal processing frameworks, potentially leading to hybrid models that are both interpretable and high-performing.

In conclusion, this paper builds a foundationally sound approach for noisy sparse phase retrieval, providing a pathway forward for deploying statistical and computational synergy in addressing complex inverse problems within signal processing and beyond.