Weakly monotone averaging functions

TL;DR

This paper introduces the concept of weakly monotonic averaging functions, broadening the theoretical framework to include important non-monotonic means and establishing conditions for their weak monotonicity.

Contribution

It defines weak monotonicity for averaging functions, studies their properties, and proves several important non-monotonic means are weakly monotonic, connecting monotonic and non-monotonic aggregation.

Findings

Several non-monotonic means are weakly monotonic.

Conditions for weak monotonicity of Lehmer mean and mixture operators.

Weak monotonicity of spatial-tonal filters like bilateral filter.

Abstract

Monotonicity with respect to all arguments is fundamental to the definition of aggregation functions. It is also a limiting property that results in many important non-monotonic averaging functions being excluded from the theoretical framework. This work proposes a definition for weakly monotonic averaging functions, studies some properties of this class of functions and proves that several families of important non-monotonic means are actually weakly monotonic averaging functions. Specifically we provide sufficient conditions for weak monotonicity of the Lehmer mean and generalised mixture operators. We establish weak monotonicity of several robust estimators of location and conditions for weak monotonicity of a large class of penalty-based aggregation functions. These results permit a proof of the weak monotonicity of the class of spatial-tonal filters that include important members such as the bilateral filter and anisotropic diffusion. Our concept of weak monotonicity provides a sound theoretical and practical basis by which (monotone) aggregation functions and non-monotone averaging functions can be related within the same framework, allowing us to bridge the gap between these previously disparate areas of research.