  Consider reconstructing a signal $x$ by minimizing a weighted sum of a convex
differentiable negative log-likelihood (NLL) (data-fidelity) term and a convex
regularization term that imposes a convex-set constraint on $x$ and enforces
its sparsity using $\ell_1$-norm analysis regularization. We compute upper
bounds on the regularization tuning constant beyond which the regularization
term overwhelmingly dominates the NLL term so that the set of minimum points of
the objective function does not change. Necessary and sufficient conditions for
irrelevance of sparse signal regularization and a condition for the existence
of finite upper bounds are established. We formulate an optimization problem
for finding these bounds when the regularization term can be globally minimized
by a feasible $x$ and also develop an alternating direction method of
multipliers (ADMM) type method for their computation. Simulation examples show
that the derived and empirical bounds match.
