  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.
