Approximations of Lovasz extensions and their induced interaction index

TL;DR

This paper studies how to best approximate Lovasz extensions of pseudo-Boolean functions with lower-degree extensions using least squares, introduces a new interaction index from these approximations, and relates it to existing indices.

Contribution

It provides explicit formulas for least squares approximations of Lovasz extensions and introduces a new interaction index derived from these approximations, linking it to known power indices.

Findings

Explicit expressions for approximations of Lovasz extensions.

Introduction of a new interaction index from approximations.

The new index coincides with a known power index.

Abstract

The Lovasz extension of a pseudo-Boolean function $f : \{0,1\}^n \to R$ is defined on each simplex of the standard triangulation of $[0,1]^n$ as the unique affine function $\hat f : [0,1]^n \to R$ that interpolates $f$ at the $n+1$ vertices of the simplex. Its degree is that of the unique multilinear polynomial that expresses $f$. In this paper we investigate the least squares approximation problem of an arbitrary Lovasz extension $\hat f$ by Lovasz extensions of (at most) a specified degree. We derive explicit expressions of these approximations. The corresponding approximation problem for pseudo-Boolean functions was investigated by Hammer and Holzman (1992) and then solved explicitly by Grabisch, Marichal, and Roubens (2000), giving rise to an alternative definition of Banzhaf interaction index. Similarly we introduce a new interaction index from approximations of $\hat f$ and we present some of its properties. It turns out that its corresponding power index identifies with the power index introduced by Grabisch and Labreuche (2001).