  We present an efficient mathematical framework based on the linearly-involved
Moreau-enhanced-over-subspace (LiMES) model. Two concrete applications are
considered: sparse modeling and robust regression. The popular minimax concave
(MC) penalty for sparse modeling subtracts, from the $\ell_1$ norm, its Moreau
envelope, inducing nearly unbiased estimates and thus yielding remarkable
performance enhancements. To extend it to underdetermined linear systems, we
propose the projective minimax concave penalty using the projection onto the
input subspace, where the Moreau-enhancement effect is restricted to the
subspace for preserving the overall convexity. We also present a novel concept
of stable outlier-robust regression which distinguishes noise and outlier
explicitly. The LiMES model encompasses those two specific examples as well as
two other applications: stable principal component pursuit and robust
classification. The LiMES function involved in the model is an ``additively
nonseparable'' weakly convex function but is defined with the Moreau envelope
returning the minimum of a ``separable'' convex function. This mixed nature of
separability and nonseparability allows an application of the LiMES model to
the underdetermined case with an efficient algorithmic implementation. Two
linear/affine operators play key roles in the model: one corresponds to the
projection mentioned above and the other takes care of robust
regression/classification. A necessary and sufficient condition for convexity
of the smooth part of the objective function is studied. Numerical examples
show the efficacy of LiMES in applications to sparse modeling and robust
regression.
