The iterative shrinkage thresholding (IST) is one of the most effective methods to solve problem 2:
denotes the coefficients model after th iteration,
denotes a thresholding operator (which is a nonlinear operator) with an input parameter , and denotes the adjoint operator. It's worth noting that has different connections with according to the thresholding type (Chen et al., 2014a).
Due to the convergence of IST, implementing IST is usually very time-consuming in practice. Beck and Teboulle (2009) proposed the fast iterative shrinkage thresholding algorithm (FISTA) to improve the convergence rate:
where is a controlling parameter with the initial value and
The improved convergence rate is , and thus becomes widely used in the image-processing field since its invention.