## Tags : mean

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## Jan 11, 2011

### Lagrangean and Quasi-arithmetic means

Post @ 15:37:15 | mean

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.

## Oct 28, 2010

### Mean/Centroid of rank-deficient symmetric positive definite matrices (SPDs)

Post @ 18:20:43 | Riemannian mean

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.

Frank.