Data-Time Tradeoffs for Optimal k-Thresholding Algorithms in Compressed Sensing

TL;DR

This paper analyzes the convergence and data-time tradeoffs of optimal k-thresholding algorithms in compressed sensing, showing how measurement quantity affects recovery success and convergence speed.

Contribution

It provides a novel convergence analysis revealing the data-time tradeoffs and measurement requirements for optimal k-thresholding algorithms in compressed sensing.

Findings

Algorithms fail to converge with few measurements.

More measurements lead to faster, linear convergence.

Measurement count of order k log(n/k) suffices for successful recovery.

Abstract

Optimal $k$-thresholding algorithms are a class of $k$-sparse signal recovery algorithms that overcome the shortcomings of traditional hard thresholding algorithms caused by the oscillation of the residual function. In this paper, a novel convergence analysis for optimal $k$-thresholding algorithms is established, which reveals the data-time tradeoffs of these algorithms. Both the analysis and numerical results demonstrate that when the number of measurements is small, the algorithms cannot converge; when the number of measurements is suitably large, the number of iterations required for successful recovery has a negative correlation with the number of measurements, and the algorithms can achieve linear convergence. Furthermore, the main theorems indicate that the number of measurements required for successful recovery is of the order of $k \log({n}/{k})$, where $n$ is the dimension of the target signal.