FAASTA: A fast solver for total-variation regularization of ill-conditioned problems with application to brain imaging
This work addresses the computational bottleneck in ill-conditioned inverse problems like brain imaging, offering a faster method for researchers in medical imaging and signal processing, though it is incremental as it builds on existing FISTA techniques.
The authors tackled the challenge of solving total-variation regularization problems, which are non-smooth and ill-conditioned, by developing FAASTA, a fast solver that balances computational costs between gradient and proximal steps. They demonstrated its effectiveness in brain imaging applications, showing improved convergence speed in benchmarks for brain decoding tasks.
The total variation (TV) penalty, as many other analysis-sparsity problems, does not lead to separable factors or a proximal operatorwith a closed-form expression, such as soft thresholding for the $\ell\_1$ penalty. As a result, in a variational formulation of an inverse problem or statisticallearning estimation, it leads to challenging non-smooth optimization problemsthat are often solved with elaborate single-step first-order methods. When thedata-fit term arises from empirical measurements, as in brain imaging, it isoften very ill-conditioned and without simple structure. In this situation, in proximal splitting methods, the computation cost of thegradient step can easily dominate each iteration. Thus it is beneficialto minimize the number of gradient steps.We present fAASTA, a variant of FISTA, that relies on an internal solver forthe TV proximal operator, and refines its tolerance to balance computationalcost of the gradient and the proximal steps. We give benchmarks andillustrations on "brain decoding": recovering brain maps from noisymeasurements to predict observed behavior. The algorithm as well as theempirical study of convergence speed are valuable for any non-exact proximaloperator, in particular analysis-sparsity problems.