  $\ell_1$ mean filtering is a conventional, optimization-based method to
estimate the positions of jumps in a piecewise constant signal perturbed by
additive noise. In this method, the $\ell_1$ norm penalizes sparsity of the
first-order derivative of the signal. Theoretical results, however, show that
in some situations, which can occur frequently in practice, even when the jump
amplitudes tend to $\infty$, the conventional method identifies false change
points. This issue is referred to as stair-casing problem and restricts
practical importance of $\ell_1$ mean filtering. In this paper, sparsity is
penalized more tightly than the $\ell_1$ norm by exploiting a certain class of
nonconvex functions, while the strict convexity of the consequent optimization
problem is preserved. This results in a higher performance in detecting change
points. To theoretically justify the performance improvements over $\ell_1$
mean filtering, deterministic and stochastic sufficient conditions for exact
change point recovery are derived. In particular, theoretical results show that
in the stair-casing problem, our approach might be able to exclude the false
change points, while $\ell_1$ mean filtering may fail. A number of numerical
simulations assist to show superiority of our method over $\ell_1$ mean
filtering and another state-of-the-art algorithm that promotes sparsity tighter
than the $\ell_1$ norm. Specifically, it is shown that our approach can
consistently detect change points when the jump amplitudes become sufficiently
large, while the two other competitors cannot.
