Download Matrix Information Geometry by Jesús Angulo (auth.), Frank Nielsen, Rajendra Bhatia (eds.) PDF
By Jesús Angulo (auth.), Frank Nielsen, Rajendra Bhatia (eds.)
This booklet offers advances in matrix and tensor facts processing within the area of sign, photograph and data processing. The theoretical mathematical ways are discusses within the context of strength functions in sensor and cognitive structures engineering.
The subject matters and alertness comprise details Geometry, Differential Geometry of established Matrix, optimistic convinced Matrix, Covariance Matrix, Sensors (Electromagnetic Fields, Acoustic sensors) and functions in Cognitive platforms, specifically information Mining.
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Extra info for Matrix Information Geometry
Example text
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.