  In this paper, we propose a new framework for solving state estimation
problems with an additional sparsity-promoting $L_1$-regularizer term. We first
formulate such problems as minimization of the sum of linear or nonlinear
quadratic error terms and an extra regularizer, and then present novel
algorithms which solve the linear and nonlinear cases. The methods are based on
a combination of the iterated extended Kalman smoother and variable splitting
techniques such as alternating direction method of multipliers (ADMM). We
present a general algorithmic framework for variable splitting methods, where
the iterative steps involving minimization of the nonlinear quadratic terms can
be computed efficiently by iterated smoothing. Due to the use of state
estimation algorithms, the proposed framework has a low per-iteration time
complexity, which makes it suitable for solving a large-scale or
high-dimensional state estimation problem. We also provide convergence results
for the proposed algorithms. The experiments show the promising performance and
speed-ups provided by the methods.
