Total Variation (TV) is a popular regularization strategy that promotes piece-wise constant signals by constraining the ℓ1-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.

Summary and Contributions: The paper considers the problem of accelerating TV-regularized problems. The paper first shows that assuming random Gaussian design, the analysis formulation might converge faster than the synthesis formulation based on the usual convergence rate for PGD. The paper then proposes two methods to accelerate the analysis formulation, using unrolling as done in LISTA. With regular lasso, the proximal operator is soft thresholding and back propagation is easy. With TV, backpropagation is a bit more difficult, and the paper proposes two alternatives.

Total Variation (TV) is a popular regularization strategy that promotes piece-wise constant signals by constraining the ℓ1-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.