Download Methods of Information Geometry (Translations of by Shun-Ichi Amari, Hiroshi Nagaoka PDF
By Shun-Ichi Amari, Hiroshi Nagaoka
Read or Download Methods of Information Geometry (Translations of Mathematical Monographs 191) PDF
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Given m points a1 , . . , am in a Euclidean space, the function m x − aj 2 j=1 has a unique minimum at a¯ = 1 (a1 + · · · + am ), m 2 The Riemannian Mean of Positive Matrices 43 the arithmetic mean of a1 , . . , am . This is the “Euclidean barycentre” of these points. When m = 3, the point a¯ is the point where the three medians of the triangle with vertices a1 , a2 , a3 intersect. This is also the point that lies in the intersection of the nested sequence of triangles { k } obtained by the procedure outlined at the end of Sect.
In F. Nielsen and R. 1007/978-3-642-30232-9_2, © Springer-Verlag Berlin Heidelberg 2013 35 36 R. Bhatia (v) m is continuous. Other requirements may be imposed, if needed, in a particular context. 1) , 2 2 respectively. 2) 0 much used in heat flow problems; and the binomial means defined as B p (a, b) = ap + bp 2 1/ p , −∞ < p < ∞. 6) p→0 p→∞ p→−∞ are also means. In various contexts we wish to have a notion of a mean of two positive definite (positive, for short) matrices. Several interesting problems arise.
Riemannian geometry and matrix geometric means. Linear Algebra Appl. 413, 594–618 (2006) 12. : Monotonicity of the matrix geometric mean, to appear in Math. Ann. (2011) 13. : Computing the Karcher mean of symmetric positive definite matrices, to appear in Linear Algebra Appl. (2012) 14. : An effective matrix geometric mean satisfying the Ando-LiMathias properties. Math. Comp. 79, 437–452 (2010) 15. : Circuit properties of coupled dispersive lines with applications to wave guide modelling. O. ) Proceedings on Network and Signal Theory, pp.