[Paper Review] SPRSF: Sparse Phase Retrieval via Smoothing Function

Samuel Pinilla, Jorge Bacca+1 more|arXiv (Cornell University)|Jul 25, 2018

Advanced X-ray Imaging Techniques15 references4 citations

TL;DR

This paper proposes SPRSF, a novel sparse phase retrieval algorithm that uses a smoothing function to overcome the non-smoothness of existing methods like SPARTA, eliminating the need for truncation thresholds. SPRSF employs iterative hard thresholding on a gradient descent direction and achieves linear convergence with sampling complexity O(k² log n), outperforming state-of-the-art methods in recovery rate and robustness, especially when sparsity k is unknown.

ABSTRACT

Phase retrieval (PR) is an ill-conditioned inverse problem which can be found in various science and engineering applications. Assuming sparse priority over the signal of interest, recent algorithms have been developed to solve the phase retrieval problem. Some examples include SparseAltMinPhase (SAMP), Sparse Wirtinger flow (SWF) and Sparse Truncated Amplitude flow (SPARTA). However, the optimization cost functions of the mentioned algorithms are non-convex and non-smooth. In order to fix the non-smoothness of the cost function, the SPARTA method uses truncation thresholds to calculate a truncated step update direction. In practice, the truncation procedure requires calculating more parameters to obtain a desired performance in the phase recovery. Therefore, this paper proposes an algorithm called SPRSF (Sparse Phase retrieval via Smoothing Function) to solve the sparse PR problem by introducing a smoothing function. SPRSF is an iterative algorithm where the update step is obtained by a hard thresholding over a gradient descent direction. Theoretical analyses show that the smoothing function uniformly approximates the non-convex and non-smooth sparse PR optimization problem. Moreover, SPRSF does not require the truncation procedure used in SPARTA. Numerical tests demonstrate that SPRSF performs better than state-of-the-art methods, especially when there is no knowledge about the sparsity $k$. In particular, SPRSF attains a higher mean recovery rate in comparison with SPARTA, SAMP and SWF methods, when the sparsity varies for the real and complex cases. Further, in terms of the sampling complexity, the SPRSF method outperforms its competitive alternatives.

Motivation & Objective

• To address the non-smoothness and high sampling complexity of existing sparse phase retrieval algorithms such as SPARTA, SAMP, and SWF.
• To develop a smooth, non-convex optimization framework that uniformly approximates the sparse phase retrieval problem.
• To eliminate the need for truncation thresholds in gradient updates, which otherwise require tuning and degrade performance.
• To achieve linear convergence with minimal sampling complexity for k-sparse signals.
• To improve recovery performance, especially when the true sparsity level k is unknown.

Proposed method

• Introduces a smoothing function to approximate the non-smooth, non-convex cost function in sparse phase retrieval.
• Uses a gradient descent direction updated via iterative hard thresholding to maintain sparsity.
• Employs a smooth approximation of the absolute value and phase terms in the cost function to ensure differentiability.
• Derives theoretical bounds on the gradient norm using spectral norms and concentration inequalities.
• Establishes convergence under the condition that the measurement matrix satisfies restricted isometry properties with high probability.
• Proves linear convergence for any k-sparse signal with sampling complexity O(k² log n).

Experimental results

Research questions

• RQ1 Can a smoothing function effectively approximate the non-smooth, non-convex sparse phase retrieval cost function?
• RQ2 Does eliminating truncation thresholds in gradient updates lead to improved recovery performance and reduced sampling complexity?
• RQ3 Can SPRSF achieve linear convergence without requiring prior knowledge of the sparsity level k?
• RQ4 How does SPRSF compare to SPARTA, SAMP, and SWF in terms of recovery rate and robustness under varying sparsity?
• RQ5 What is the theoretical sampling complexity of SPRSF for exact signal recovery?

Key findings

• SPRSF achieves linear convergence for any k-sparse signal with sampling complexity O(k² log n), matching theoretical bounds.
• SPRSF outperforms SPARTA, SAMP, and SWF in mean recovery rate, especially when the sparsity level k is unknown.
• The algorithm eliminates the need for truncation thresholds, reducing parameter tuning and improving stability.
• Numerical results show SPRSF maintains high recovery performance across both real and complex signal cases under varying sparsity.
• Theoretical analysis confirms that the smoothing function uniformly approximates the original non-smooth cost function.
• SPRSF demonstrates robustness to uncertainty in sparsity, making it suitable for practical applications where k is not known a priori.