Computational Information Geometry Wonderland

First of all, happy new year and best wishes to all Readers. In Asia, 2011 is a Rabbit year.

The Lagrangean mean $\bar L_f(p,q)$ of two reals $p$ and $q$ is defined for a continuous and strictly monotonic function $f$ as

$$ \bar L_f(p,q)= f^{-1} (\frac{1}{q-p} \int_p^q f(x) \mathrm{d}x ) $$ if $p\not =q$, or $\bar L_f(p,q)==p$ if $p=q$.

It is related to the mean value theorem

$$ f'(m)=\frac{f(q)-f(p)}{q-p}=\frac{\int_p^q f'(x) \mathrm{d}x}{q-p} $$

The Lagrangean mean is symmetric and is defined up to some affine term $ax+b$ as quasi-arithmetic means $$ Q_f(p,q)=f^{-1}\ (\frac{f(p)+f(q)}{2} ) $$

$Q_f=L_f$ only for $f(x)=x$, the arithmetic mean, otherwise the means differ.

Here is a reference :
Justyna Jarczyk
When lagrangean and quasi-arithmetic means coincide
JOURNAL OF INEQUALITIES IN PURE AND APPLIED MATHEMATICS
2007, Vol. 8, no 3

Frank.

Geometry of matrix manifolds is fascinating. (I will give a talk on this topic next December) A very recent and neat work generalizes the Riemannian mean of (SPD) positive semi-definite matrices to rank deficient sets using quotient geometry:

Rank-preserving geometric means of positive semi-definite matrices
by Silvere Bonnabel, Rodolphe Sepulchre.

http://arxiv.org/abs/1007.5494

Frank.