A Differential Geometric Approach to the Geometric Mean of Symmetric Positive-Definite Matrices

By Maher Moakher. SIAM Journal on Matrix Analysis and Applications, 2005

769 citations recorded by the source corpus.

In this paper we introduce metric-based means for the space of positive-definite matrices. The mean associated with the Euclidean metric of the ambient space is the usual arithmetic mean. The mean associated with the Riemannian metric corresponds to the geometric mean. We discuss some invariance properties of the Riemannian mean and we use differential geometric tools to give a characterization of this mean.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/ec6ac4fd164f31ab9274c8a3791128bfb33f3682

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Astronomy and Astrophysics

In Computational Theory and Mathematics

Citations in the corpus

Classification of covariance matrices using a Riemannian-based kernel for BCI applications (2013, methodology): “Thefullalgorithm, derived from Moakher (2005), is given by the algorithm 1.”
Riemannian Approaches in Brain-Computer Interfaces: A Review (2017, neutral): “Such an averaging method has attracted a lot of attention from the optimization community and for example, for SPD matrices, several algorithms have been proposed [34]–[36].”
Multiclass Brain–Computer Interface Classification by Riemannian Geometry (2012, background): “Such mean is also referred to as the arithmetic mean in the literature [13] and it has been used ubiquitously so far.”
Transfer Learning: A Riemannian Geometry Framework With Applications to Brain–Computer Interfaces (2018, background): “Since P (n) is a Riemannian manifold of non-positive curvature, existence and unicity of the Riemannian mean can be proved [1], [28].”
On the Averaging of Symmetric Positive-Definite Tensors (2006, background): “This mean is in fact given by the polar factor in the polar decomposition of the arithmetic mean of the orthogonal matrices of eigenvectors [6, 18 ].”
Matrix power means and the Karcher mean (2012, background): “1 of [18]) or computing appropriate derivatives as in [10,28] yields that the Karcher mean coincides with the unique positive definite solution of the Karcher equation”
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].”
Monotonic properties of the least squares mean (2010, background): “In their study of the unweighted least squares mean, Moakher [ 23 ] and Bhatia and”
Karcher means and Karcher equations of positive definite operators (2014, methodology): “Then, via the Karcher equation, the Karcher mean satisfies the joint homogeneity property ([18]) Λ(ω; a1A1, .”
Pedestrian Detection via Classification on Riemannian Manifolds (2008, background): “However, it is shown in several studies that the mean is unique, and the gradient descent algorithm is convergent for Symd [13], [29], [35].”
Embedded geometry of the set of symmetric positive semidefinite matrices of fixed rank (2009, background): “To derive this ODE for S(t) such that (8)–(9) are satisfied, we make the substitution S(t) = Y (t)Y (t) with Y (t) ∈ R ∗ and derive an ODE directly for Y (t).”
Geometry of Logarithmic Strain Measures in Solid Mechanics (2015, methodology): “An explicit formula for the corresponding geodesic distance was given by Moakher:32 distgeod, Sym+(n)(C1, C2) = ‖log(C−1/22 C1 C −1/2 2 )‖ .”
The Closest Elastic Tensor of Arbitrary Symmetry to an Elasticity Tensor of Lower Symmetry (2006, background): “The midpoint with dR is [24] C3 = C1( C−1 1 C2) which can be expressed in several other ways, see [1].”