Computationally Efficient Sparse Signal Recovery via Linear Sketching and Deep Unfolding

TL;DR

This paper introduces DU-PSISTA, a deep unfolded sparse signal recovery algorithm that combines sketching and iterative thresholding to improve efficiency and accuracy, with theoretical convergence guarantees and practical performance benefits.

Contribution

The paper proposes a novel hybrid deep unfolded algorithm using sketching and periodic ISTA updates, balancing computational efficiency and recovery accuracy.

Findings

Achieves linear contraction to the true sparse signal with proper parameters.

Reduces computational complexity compared to conventional deep unfolded ISTA.

Experimental results confirm comparable recovery performance with improved efficiency.

Abstract

This paper provides a sparse signal recovery algorithm, DU-PSISTA (Deep Unfolded-Periodic Sketched Iterative Shrinkage-Thresholding Algorithm), which aims to balance computational efficiency and accuracy for recovering high-dimensional sparse signals, and a convergence analysis under sufficient conditions. DU-PSISTA introduces a random matrix projection known as sketching to reduce the dimensionality of gradient computations and periodically alternates between the standard ISTA and the sketched variant. This hybrid structure enables flexible control over the trade-off between accuracy and computational complexity through a pre-configurable period parameter. The algorithm includes many parameters to be tuned such as step sizes and thresholding factors so that we incorporate deep unfolding that optimizes the parameters through data-driven training, enabling the algorithm to adaptively improve convergence speed and performance. We show that the proposed method achieves a linear-type contraction to a neighborhood of the true sparse signal with properly selected parameters. The analysis provides an interpretation for the effectiveness of the hybrid structure to improve recovery accuracy. Numerical experiments confirm that our method achieves comparable recovery performance to conventional deep unfolded ISTA while reducing computational complexity, especially when the period parameter and sketch size are properly selected. The results are also consistent with the theoretical insights.