A Class of Nonconvex Penalties Preserving Overall Convexity in Optimization-Based Mean Filtering

TL;DR

This paper introduces a class of nonconvex penalties that preserve overall convexity in optimization-based mean filtering, significantly improving change point detection accuracy over traditional  methods, especially in stair-casing scenarios.

Contribution

The authors propose a novel nonconvex penalty framework that maintains convexity, leading to better detection of true change points and exclusion of false ones compared to  mean filtering.

Findings

The method outperforms  mean filtering in detecting large amplitude jumps.

The approach effectively excludes false change points in stair-casing problems.

Numerical simulations demonstrate superior performance over existing algorithms.

Abstract

$\ell_1$ 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.