Heinz mean
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Short description: Mean in mathematics
In mathematics, the Heinz mean (named after E. Heinz[1]) of two non-negative real numbers A and B, was defined by Bhatia[2] as:
- [math]\displaystyle{ \operatorname{H}_x(A, B) = \frac{A^x B^{1-x} + A^{1-x} B^x}{2}, }[/math]
with 0 ≤ x ≤ 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 < 1/2:
- [math]\displaystyle{ \sqrt{A B} = \operatorname{H}_\frac{1}{2}(A, B) \lt \operatorname{H}_x(A, B) \lt \operatorname{H}_0(A, B) = \frac{A + B}{2}. }[/math]
The Heinz means appear naturally when symmetrizing [math]\displaystyle{ \alpha }[/math]-divergences.[3]
It may also be defined in the same way for positive semidefinite matrices, and satisfies a similar interpolation formula.[4][5]
See also
References
- ↑ 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, doi:10.3390/e16063273, Bibcode: 2014Entrp..16.3273N.
- ↑ 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, doi:10.1016/j.laa.2006.10.006.
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