What is an inductive mean?

TL;DR

This paper explores the concept of inductive means, generalizing the scalar geometric mean to symmetric positive-definite matrices and presenting mechanisms for their inductive computation.

Contribution

It introduces new inductive mean mechanisms for symmetric positive-definite matrices, extending classical scalar concepts to matrix settings.

Findings

Generalizations of scalar geometric mean to matrices

Multiple inductive mean mechanisms for matrices

Connections to classical inductive mean concepts

Abstract

An inductive mean is a mean defined as a limit of a convergence sequence of other means. Historically, this notion of inductive means obtained as limits of sequences was pioneered independently by Lagrange and Gauss for defining the arithmetic-geometric mean. In this note, we first explain several generalizations of the scalar geometric mean to symmetric positive-definite matrices, and then present several inductive mean mechanisms for sets of symmetric positive-definite matrices.