A new perspective on the Propagation-Separation approach: Taking advantage of the propagation condition

TL;DR

This paper offers a new perspective on the Propagation-Separation method by analyzing a simplified version without the memory step, introducing a novel strategy for the adaptation parameter to improve stability and propagation in local constant functions with sharp discontinuities.

Contribution

It provides a theoretical analysis of the simplified Propagation-Separation algorithm and proposes a new strategy for selecting the adaptation parameter to enhance performance.

Findings

The simplified algorithm maintains stability without the memory step.

A new adaptation parameter strategy improves propagation in discontinuous regions.

Theoretical guarantees are established for local constant functions with sharp edges.

Abstract

The Propagation-Separation approach is an iterative procedure for pointwise estimation of local constant and local polynomial functions. The estimator is defined as a weighted mean of the observations with data-driven weights. Within homogeneous regions it ensures a similar behavior as non-adaptive smoothing (propagation), while avoiding smoothing among distinct regions (separation). In order to enable a proof of stability of estimates, the authors of the original study introduced an additional memory step aggregating the estimators of the successive iteration steps. Here, we study theoretical properties of the simplified algorithm, where the memory step is omitted. In particular, we introduce a new strategy for the choice of the adaptation parameter yielding propagation and stability for local constant functions with sharp discontinuities.