  We consider a minimization problem whose objective function is the sum of a
fidelity term, not necessarily convex, and a regularization term defined by a
positive regularization parameter $\lambda$ multiple of the $\ell_0$ norm
composed with a linear transform. This problem has wide applications in
compressed sensing, sparse machine learning and image reconstruction. The goal
of this paper is to understand what choices of the regularization parameter can
dictate the level of sparsity under the transform for a global minimizer of the
resulting regularized objective function. This is a critical issue but it has
been left unaddressed. We address it from a geometric viewpoint with which the
sparsity partition of the image space of the transform is introduced. Choices
of the regularization parameter are specified to ensure that a global minimizer
of the corresponding regularized objective function achieves a prescribed level
of sparsity under the transform. Results are obtained for the spacial sparsity
case in which the transform is the identity map, a case that covers several
applications of practical importance, including machine learning, image/signal
processing and medical image reconstruction.
