Monotone FISTA with Variable Acceleration for Compressed Sensing Magnetic Resonance Imaging

An improvement of the monotone fast iterative shrinkage-thresholding algorithm (MFISTA) for faster convergence is proposed. Our motivation is to reduce the reconstruction time of compressed sensing problems in magnetic resonance imaging. The proposed modification introduces an extra term, which is a multiple of the proximal-gradient step, into the so-called momentum formula used for the computation of the next iterate in MFISTA. In addition, the modified algorithm selects the next iterate as a possibly-improved point obtained by any other procedure, such as an arbitrary shift, a line search, or other methods. As an example, an arbitrary-length shift in the direction from the previous iterate to the output of the proximal-gradient step is considered. The resulting algorithm accelerates MFISTA in a manner that varies with the iterative steps. Convergence analysis shows that the proposed modification provides improved theoretical convergence bounds, and that it has more flexibility in its parameters than the original MFISTA. Since such problems need to be studied in the context of functions of several complex variables, a careful extension of FISTA-like methods to complex variables is provided.