penalized least-squares minimization
The area of sparse representation of signals is drawing tremendous
attention in recent years. The idea behind the model is that a
signal can be approximated as a linear combination of a few
``atoms'' from a prespecified and over-complete ``dictionary''.
The sparse representation of a signal is often achieved by
minimizing an 
penalized least squares functional. Various
iterative-shrinkage algorithms have recently been developed and
are quite effective for handling these problems, surpassing
traditional optimization techniques. In this paper, we suggest a
new simple multilevel approach that reduces the computational cost
of existing solvers for these inverse problems. The new method
takes advantage of the typically sparse representation of the
signal and coarsens by reducing the dimension of the problem by
gradually ignoring ostensibly irrelevant data from the
over-complete dictionary. Analytical observations suggest, and
numerical results confirm, that this new approach may
significantly enhance the performance of existing iterative
shrinkage algorithms in cases where the dictionary is an explicit
matrix.