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Heinz mean

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Mean in mathematics

In mathematics, theHeinz mean (named afterE. Heinz^[1]) of two non-negativereal numbersA andB, was defined by Bhatia^[2] as:

\operatorname {H} _{x}(A,B)={\frac {A^{x}B^{1-x}+A^{1-x}B^{x}}{2}},

with 0 ≤ x ≤ ⁠1/2⁠.

For different values ofx, this Heinz mean interpolates between thearithmetic (x = 0) andgeometric (x = 1/2) means such that for 0 < x < ⁠1/2⁠:

{\sqrt {AB}}=\operatorname {H} _{\frac {1}{2}}(A,B)<\operatorname {H} _{x}(A,B)<\operatorname {H} _{0}(A,B)={\frac {A+B}{2}}.

The Heinz means appear naturally when symmetrizing ${\textstyle \alpha }$ -divergences.^[3]

It may also be defined in the same way forpositive semidefinite matrices, and satisfies a similar interpolation formula.^[4]^[5]

^E. Heinz (1951), "Beiträge zur Störungstheorie der Spektralzerlegung",Math. Ann.,123, pp. 415–438.
^Bhatia, R. (2006), "Interpolating the arithmetic-geometric mean inequality and its operator version",Linear Algebra and Its Applications,413 (2–3):355–363,doi:10.1016/j.laa.2005.03.005.
^Nielsen, Frank; Nock, Richard; Amari, Shun-ichi (2014), "On Clustering Histograms with k-Means by Using Mixed α-Divergences",Entropy,16 (6):3273–3301,Bibcode:2014Entrp..16.3273N,doi:10.3390/e16063273,hdl:1885/98885.
^Bhatia, R.;Davis, C. (1993), "More matrix forms of the arithmetic-geometric mean inequality",SIAM Journal on Matrix Analysis and Applications,14 (1):132–136,doi:10.1137/0614012.
^Audenaert, Koenraad M.R. (2007), "A singular value inequality for Heinz means",Linear Algebra and Its Applications,422 (1):279–283,arXiv:math/0609130,doi:10.1016/j.laa.2006.10.006,S2CID 15032884.

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