Iterative Thresholding Algorithms for Compressive Sensing: Explained

Iterative thresholding algorithms are a family of methods used in compressive sensing to solve the linear inverse problem of reconstructing a sparse signal from a small number of linear measurements. The basic idea of these algorithms is to iteratively update the estimate of the sparse signal by thresholding the measurements in a way that promotes sparsity.

The most common iterative thresholding algorithms are:

1. Iterative Soft Thresholding (IST): This algorithm updates the estimate of the sparse signal by applying a soft thresholding operator to the measurements. The soft thresholding operator shrinks the measurements towards zero and encourages sparsity in the estimate.

2. Iterative Hard Thresholding (IHT): This algorithm updates the estimate of the sparse signal by selecting the largest k coefficients (where k is the sparsity level) and setting the rest to zero. This promotes sparsity by forcing the estimate to have only k non-zero coefficients.

3. Iterative Group Thresholding (IGT): This algorithm updates the estimate of the sparse signal by thresholding groups of coefficients simultaneously. This is useful when the sparse signal has a structured sparsity pattern, such as in image or video data.

These iterative thresholding algorithms have been shown to be effective in a wide range of applications, including image compression, signal denoising, and machine learning.