  We consider a regularization problem whose objective function consists of a
convex fidelity term and a regularization term determined by the $\ell_1$ norm
composed with a linear transform. Empirical results show that the
regularization with the $\ell_1$ norm can promote sparsity of a regularized
solution. It is the goal of this paper to understand theoretically the effect
of the regularization parameter on the sparsity of the regularized solutions.
We establish a characterization of the sparsity under the transform matrix of
the solution. When the fidelity term has a special structure and the transform
matrix coincides with a identity matrix, the resulting characterization can be
taken as a regularization parameter choice strategy with which the
regularization problem has a solution having a sparsity of a certain level. We
study choices of the regularization parameter so that the regularization term
alleviates the ill-posedness and promote sparsity of the resulting regularized
solution. Numerical experiments demonstrate that choices of the regularization
parameters can balance the sparsity of the solutions of the regularization
problem and its approximation to the minimizer of the fidelity function.
