A Note on Affine Invariant Cost Functions

TL;DR

This paper proves that affine invariant functions on positive definite matrices are essentially determined by the determinant, under certain conditions, with implications for robust statistics.

Contribution

It establishes a fundamental characterization of affine invariant functions on positive definite matrices, linking them to the determinant function.

Findings

Affine invariant functions factor through the determinant.

The result applies when the function's restriction to scalar matrices is surjective.

Implications discussed for robust statistical methods.

Abstract

We show that any affine invariant function on the set of positive definite matrices must factor through the determinant function, as long as the restriction of the function to scalar matrices is surjective. A motivation from robust statistics is discussed.