A Class of Nonconvex Penalties Preserving Overall Convexity in Optimization-Based Mean Filtering
ℓ1 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 ℓ1 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 ∞, the conventional method identifies false change points. This issue, which is referred to as the stair-casing problem herein, restricts practical importance of ℓ1 mean filtering. In this paper, sparsity is penalized more tightly than the ℓ1 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 ℓ1 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 ℓ1 mean filtering may fail. A number of numerical simulations assist to show superiority of our method over ℓ1 mean filtering and another state-of-the-art algorithm that promotes sparsity tighter than the ℓ1 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.