Learning to solve TV regularized problems with unrolled algorithms
This work addresses the computational bottleneck in solving TV regularized optimization problems, which is incremental as it builds on existing unrolling techniques for a specific domain.
The paper tackles the slow convergence of iterative algorithms for Total Variation (TV) regularized problems by learning parameters through unrolled proximal gradient descent solvers, demonstrating improved performance over traditional methods in experiments on synthetic and real data.
Total Variation (TV) is a popular regularization strategy that promotes piece-wise constant signals by constraining the $\ell_1$-norm of the first order derivative of the estimated signal. The resulting optimization problem is usually solved using iterative algorithms such as proximal gradient descent, primal-dual algorithms or ADMM. However, such methods can require a very large number of iterations to converge to a suitable solution. In this paper, we accelerate such iterative algorithms by unfolding proximal gradient descent solvers in order to learn their parameters for 1D TV regularized problems. While this could be done using the synthesis formulation, we demonstrate that this leads to slower performances. The main difficulty in applying such methods in the analysis formulation lies in proposing a way to compute the derivatives through the proximal operator. As our main contribution, we develop and characterize two approaches to do so, describe their benefits and limitations, and discuss the regime where they can actually improve over iterative procedures. We validate those findings with experiments on synthetic and real data.