# Matrix power means and the Karcher mean

@article{Lim2012MatrixPM,
title={Matrix power means and the Karcher mean},
author={Y. Lim and Mikl{\'o}s P{\'a}lfia},
journal={Journal of Functional Analysis},
year={2012},
volume={262},
pages={1498-1514}
}
• Published 2012
• Mathematics
• Journal of Functional Analysis
We define a new family of matrix means {Pt(ω;A)}t∈[−1,1], where ω and A vary over all positive probability vectors in Rn and n-tuples of positive definite matrices resp. Each of these means except t≠0 arises as a unique positive definite solution of a non-linear matrix equation, satisfies all desirable properties of power means of positive real numbers and interpolates between the weighted harmonic and arithmetic means. The main result is that the Karcher mean coincides with the limit of power… Expand
130 Citations
The power mean and the least squares mean of probability measures on the space of positive definite matrices
• Mathematics
• 2015
Abstract In this paper we derive properties of the least squares (or Karcher) mean of probability measures on the open cone Ω of positive definite matrices of some fixed dimension endowed with theExpand
Operator means of probability measures and generalized Karcher equations
Abstract In this article we consider means of positive bounded linear operators on a Hilbert space. We present a complete theory that provides a framework which extends the theory of the KarcherExpand
Karcher means and Karcher equations of positive definite operators
• Mathematics
• 2014
The Karcher or least-squares mean has recently become an important tool for the averaging and studying of positive definite matrices. In this paper we show that this mean extends, in its generalExpand
Weighted means and Karcher equations of positive operators
• Computer Science, Medicine
• Proceedings of the National Academy of Sciences
• 2013
The Karcher or least-squares mean has recently become an important tool for the averaging and study of positive definite matrices and is extended to the infinite-dimensional setting of positive operators on a Hilbert space and retains most of its attractive properties. Expand
Strong law of large numbers for the L1-Karcher mean
• Mathematics
• 2020
Abstract Sturm's strong law of large numbers in CAT ( 0 ) spaces has been an influential tool to study the geometric mean or also called Karcher barycenter of positive definite matrices. It providesExpand
Some matrix power and Karcher means inequalities involving positive linear maps
• Mathematics
• 2018
In this paper, we generalize some matrix inequalities involving the matrix power means and Karcher mean of positive definite matrices. Among other inequalities, it is shown that if A = (A1, · · ·Expand
Strong law of large numbers for the $L^1$-Karcher mean.
• Mathematics
• 2019
Sturm's strong law of large numbers in $\mathrm{CAT}(0)$ spaces has been an influential tool to study the geometric mean or also called Karcher barycenter of positive definite matrices. It providesExpand
Fixed Point Algorithms for Estimating Power Means of Positive Definite Matrices
• Mathematics, Computer Science
• IEEE Transactions on Signal Processing
• 2017
A general fixed point algorithm (MPM) is provided and it is shown that its convergence rate for p = ±0.5 deteriorates very little with the number and dimension of points given as input, much more than the gradient descent algorithm usually employed to estimate the geometric mean. Expand
Fixed Point Algorithms for Estimating Power Means of Positive Definite Matrices
• Mathematics, Computer Science
• IEEE Trans. Signal Process.
• 2017
A general fixed point algorithm (MPM) is provided and it is shown that its convergence rate for p = ±0.5 deteriorates very little with the number and dimension of points given as input, much more than the gradient descent algorithm usually employed to estimate the geometric mean. Expand
The Karcher mean of three variables and quadric surfaces
• Mathematics
• 2020
Abstract The Riemannian or Karcher mean has recently become an important tool for the averaging and study of positive definite matrices. Finding an explicit formula for the Karcher mean isExpand

#### References

SHOWING 1-10 OF 33 REFERENCES
Multi-variable weighted geometric means of positive definite matrices
• Mathematics
• 2011
Abstract We define a family of weighted geometric means { G ( t ; ω ; A ) } t ∈ [ 0 , 1 ] n where ω and A vary over all positive probability vectors in R n and n-tuples of positive definite matricesExpand
Classification of affine matrix means
In this article we find all possible matrix means which are points of geodesics of affinely connected manifolds. We characterize certain properties of these manifolds, decide whether they areExpand
Positive Definite Matrices
This book represents the first synthesis of the considerable body of new research into positive definite matrices. These matrices play the same role in noncommutative analysis as positive realExpand
Geometric Means in a Novel Vector Space Structure on Symmetric Positive-Definite Matrices
• Mathematics, Computer Science
• SIAM J. Matrix Anal. Appl.
• 2006
This work defines the Log‐Euclidean mean from a Riemannian point of view, based on a lie group structure which is compatible with the usual algebraic properties of this matrix space and a new scalar multiplication that smoothly extends the Lie group structure into a vector space structure. Expand
Completely positive mappings and mean matrices
• Mathematics, Physics
• 2010
Abstract Some functions f : R + → R + induce mean of positive numbers and the matrix monotonicity gives a possibility for means of positive definite matrices. Moreover, such a function f can define aExpand
The Riemannian mean and matrix inequalities related to the Ando-Hiai inequality and chaotic order
The Riemannian mean on the convex cone of positive definite matrices is a kind of geometric mean of n -matrices which is an extension of the geometric mean of two-matrices. In this paper, we deriveExpand
A general framework for extending means to higher orders
• Mathematics
• 2006
Although there is an extensive literature on various means of two positive operators and their applications, these means do not typically readily extend to means of three and more operators. It hasExpand
ON CERTAIN CONTRACTION MAPPINGS IN A PARTIALLY ORDERED VECTOR SPACE
G. Birkhoff [1] and H. Samelson [4] have shown that a means of solving problems concerning the existence and uniqueness of eigenvectors of positive operators is given by introducing a suitable metricExpand
Riemannian geometry and matrix geometric means
• Mathematics
• 2006
The geometric mean of two positive definite matrices has been defined in several ways and studied by several authors, including Pusz and Woronowicz, and Ando. The characterizations by these authorsExpand
Monotonicity of the matrix geometric mean
• Mathematics
• 2012
An attractive candidate for the geometric mean of m positive definite matrices A1, . . . , Am is their Riemannian barycentre G. One of its important operator theoretic properties, monotonicity in theExpand