Riemannian geometry and matrix geometric means

By Rajendra Bhatia, J. Holbrook. 2006

301 citations recorded by the source corpus.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/9ba9dfef26a46d4fae8f12c58d4005146f837377

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In Applied Mathematics

In Computational Theory and Mathematics

In Geometry and Topology

Citations in the corpus

On the Bures–Wasserstein distance between positive definite matrices (2017, unknown): “See [7] [8] [9] [10] [19] [23] [24] [25] [26].”
Monotonic properties of the least squares mean (2010, background): “In their study of the symmetric least squares mean, Moakher [25] and Bhatia and Holbrook [7], [8] have derived for it some of the axiomatic properties (P1)-(P10) satisfied by the Ando-Li-Mathias geometric mean: consistency with scalars, joint homogeneity, permutation invariance, congruence invarianc”
Operator log-convex functions and operator means (2009, background): “(A different notion of geometric means for n operators is in [7].”
Matrix power means and the Karcher mean (2012, background): “Nevertheless the monotonicity of the Karcher mean, conjectured by Bhatia and Holbrook [11] and one of key axiomatic properties of matrix geometric means, was recently established by Lawson and Lim [24] via a probabilistic convergence of approximations and by Bhatia and Karandikar [12] via some proba”
A general framework for extending means to higher orders (2006, unknown): “Bhatia and Holbrook [6] have proposed an alternative generalization for the geometric mean via a geometric approach linked to work of É.”
Cramer-Rao Lower Bound and Information Geometry (2013, unknown): “In the case of multi-parameter distributions, the lower bound on the accuracy of unbiased estimators can be extended using the Löwner partial ordering on matrices defined by A B ⇔ A− B 0, where M 0 means M is positive semidefinite [11] (We similarly write M ≻ 0 to indicate that M is positive definit”
Computing the Karcher mean of symmetric positive definite matrices (2013, background): “This is, for instance, the definition of geometric mean of positive matrices given by Moakher [17], Bhatia and Holbrok [8].”
Generalized locally Toeplitz sequences : Theory and applications (2017, background): “, A ∈Pn was defined as the barycenter of the matrices with respect to a certain Riemannian distance; see [12] and [11, Chapter 6].”
Karcher means and Karcher equations of positive definite operators (2014, background): “First M. Moakher [18] and then Bhatia and Holbrook [5] suggested extending the geometric mean to n-points by taking the mean to be the unique minimizer of the sum of the squares Received by the editors May 18, 2012 and, in revised form, November 22, 2012.”
Geometry of Logarithmic Strain Measures in Solid Mechanics (2015, methodology): “Extensive work on the use of Lie group theory and differential geometry in continuum mechanics has already been done by Rougée [181, 180, 182, 183], Moakher [137, 139], Bhatia [26] and, more recently, by Fiala [64, 65, 66, 67, 68] (cf.”
Norm inequalities related to the matrix geometric mean (2012, unknown): “(See [4], [12] and [16].”
Positive definite matrices and the S-divergence (2011, background): “The MGM enjoys a host of attractive properties—see for instance the classic paper [1]; of these, the following variational characterization [7] is important: A B =argminX>0 δ 2 R(A,X) + δ 2 R(B,X) and δR(A,A B) = δR(B,A B).”