Heinz mean

{{Short description|Mean in mathematics}}

In mathematics, the Heinz mean (named after E. HeinzE. Heinz (1951), "Beiträge zur Störungstheorie der Spektralzerlegung", Math. Ann., 123, pp. 415–438.) of two non-negative real numbers A and B, was defined by Bhatia{{citation|first1=R.|last1=Bhatia|title=Interpolating the arithmetic-geometric mean inequality and its operator version|journal=Linear Algebra and Its Applications|volume=413|issue=2–3|pages=355–363|year=2006|doi=10.1016/j.laa.2005.03.005|doi-access=free}}. as:

:$\backslash operatorname\{H\}\_x(A,\; B)\; =\; \backslash frac\{A^x\; B^\{1-x\}\; +\; A^\{1-x\}\; B^x\}\{2\},$

with 0 ≤ x ≤ {{sfrac|1|2}}.

For different values of x, this Heinz mean interpolates between the arithmetic (x = 0) and geometric (x = 1/2) means such that for 0 < x < {{sfrac|1|2}}:

:

The Heinz means appear naturally when symmetrizing

$\backslash alpha$-divergences.

{{citation|first1=Frank |last1=Nielsen|first2=Richard|last2=Nock|first3=Shun-ichi|last3=Amari|title=On Clustering Histograms with k-Means by Using Mixed α-Divergences|journal=Entropy|year=2014|volume=16|issue=6|pages=3273–3301|doi=10.3390/e16063273|bibcode=2014Entrp..16.3273N|doi-access=free|hdl=1885/98885|hdl-access=free}}.

It may also be defined in the same way for positive semidefinite matrices, and satisfies a similar interpolation formula.{{citation|first1=R.|last1=Bhatia|first2=C.|last2=Davis|authorlink2=Chandler Davis|title=More matrix forms of the arithmetic-geometric mean inequality|journal=SIAM Journal on Matrix Analysis and Applications|volume=14|issue=1|pages=132–136|year=1993|doi=10.1137/0614012}}.{{citation|first1=Koenraad M.R.|last1=Audenaert|title=A singular value inequality for Heinz means|arxiv=math/0609130 |journal=Linear Algebra and Its Applications|volume=422|issue=1|pages=279–283|year=2007|doi=10.1016/j.laa.2006.10.006|s2cid=15032884}}.