Geometric modeling approach to the means of positive definite symmetric metrices

In this paper, we have defined and studied the geometric modeling approach to the means of positive definite symmetric metrices. In our space usual arithmetic mean is related with Euclidean metric and geometric mean is related with Riemannian metric. Also we have studied properties of Euclidean mean and invariance properties of Riemannian mean and some tools to present a classification of Riemannian metric.

Illustrations explain the paper; they do not show study data.

AI-generated illustration

↗ Enlarge

Comparison of Matrix Mean MetricsClose ×

AI-generated conceptual illustration (A curved 3D mathematical manifold surface representing the space of positive definite symmetric matrices). It explains the idea and does not show study data.

Open full-size image ↗

Introduction

Initially arithmetic, geometric and harmonic means defined for two positive numbers and then defined for a finite set of positive numbers. So for a set of m positive numbers, {𝑥 𝑘 }, 1 ≤ 𝑘 ≤ 𝑚, the arithmetic mean(always positive) is given as

For given points 𝑥 𝑘 arithmetic mean has a property that is minimizes the sum of the squared distances.

𝑥 𝑘 = arg 𝑚𝑖𝑛 𝑥 > 0 ∑ 𝑑 𝑒 (𝑥, 𝑥 𝑘 ) 2 , 𝑚 𝑘=1

Where 𝑑 𝑒 (𝑥, 𝑦) = |𝑥 -𝑦| is the usual Euclidean distance in real number system R. Geometric mean of{𝑥 𝑘 }, 1 ≤ 𝑘 ≤ 𝑚, given by 𝑥 ̃= √𝑥 1 𝑥 2 … 𝑥 𝑚 𝑚 , also it has a property that it minimizes the sum of the squared hyperbolic distances.

𝑥 ̃=

arg 𝑚𝑖𝑛 𝑥 > 0 ∑ 𝑑 ℎ (𝑥 𝑘 , 𝑥) 2 , 𝑚 𝑘=1

Where hyperbolic distance between x and y is given by 𝑑 ℎ (𝑥, 𝑦) = |𝑙𝑜𝑔𝑥 -𝑙𝑜𝑔𝑦|. As we know that harmonic mean is the inverse of the arithmetic mean of their inverses, for{𝑥 𝑘 }, 1 ≤ 𝑘 ≤ 𝑚harmonic mean 𝑥 ̂= [ ] -1 , and it has variational property as well.

Review of Literature

For regular elements of linear Euclidean spaces we generally use arithmetic mean. Arithmetic mean for application purpose referred to as the center of mass and the average. On the other hand, the geometric mean has been used only to positive numbers and positive integrable functions. In Hilbert space the geometric and harmonic means for a pair of positive operators were introduced by Anderson and Trapp, and Pusz and Woronowicz in 1975 [17] . It has been previously observed that geometric mean of two positive-definite operators holds many of the properties of geometric mean of two positive numbers. The geometric mean of positive operators has been primarily used as a binary operation.

[Trapp, G.E. (1984)] [17] have studied and investigated that, how to define the geometric mean of more than two Hermitian semidefinite matrices and used iterative method for it but it work only when matrices commute with each other. [Alice M. et al. (1997)] [1] , have given a definition for the geometric mean of a finite set of operators however, this definition was not invariant under recording of the matrices. [Moakher, M. ( 2002)] [15] , have studied about means of a finite number of 3-Dimensional rotational matrices and conclude that there is a close relation between the geometric mean of two Hermitian definite matrices and the Riemannian mean of two rotations. We assemble each the necessary background from applied differential geometry and optimization theory on manifolds that will be used in the result. [7,3,8,19] . Again, we have studied Riemannian-metric based concept of mean for two positive-definite matrices. For Riemannian mean, We have studied its invariance properties and proved that when two matrices are to be averaged, then Riemannian mean coincides with geometric mean.

Geometric modeling approach to Riemannian metric based concept of mean for positive-definite matrices Now we include circumstances, if we have Riemannian manifold μ with metric 𝑑 (𝑥, 𝑦). then by equations (1.1) and (1.2), we define mean with respect to 𝑑 (𝑥, 𝑦) for m points in μ as:

As we know for the set of positive real numbers at the same time, a Lie group and an open convex cone, different concept of mean can be correlated with different matrices. This follows that we will generalize these metric-based means to the cone positivedefinite transformations. The techniques and concepts used in this paper carry over to the complex complement of the space considered here, i.e., convex cone of Hermitian definite transformations. We here focus on the real space just for simplicity but not for any elementary reason. Let μ(n) denotes set of 𝑛 × 𝑛 real matrices and we consider its subset of all non-singular matrices denoted by GL(n). GL(n) is a group (Lie group), i.e., a group which is a differenatiable manifold and for which the operators (operations) of the group multiplication and inverse are totally valid. Lie algebra denoted by gl(n) is the tangent space at the identity. This is the space of all linear transformations in 𝑅 𝑛 , 𝑖. 𝑒. , μ(n). In μ(n), we use the Euclidean (Frobenius) inner product defined by 〈𝐴, 𝐵〉 𝐹 = 𝑡𝑟(𝐴 𝑇 𝐵), where 𝑡𝑟 denotes the trace and the superscript 𝑇 stands for the transpose. The connected norm‖𝐴‖ 𝐹 = 〈𝐴, 𝐴〉 𝐹

⁄ , is used to describe the Euclidean distance on μ(n).

The exponential of a particular matrix in 𝑔𝑙(𝑛) is given by the convergent series as:

We state that 𝐸𝑥𝑝(𝐴. 𝐵) = Exp (𝐴 + 𝐵), only when 𝐴 𝑎𝑛𝑑 𝐵 commutes with each other. In 𝐺𝐿(𝑛) solutions of the equations Exp 𝑋 = 𝐴 are Logarithms of 𝐴. When have eigenvalues in the (open) positive real line, there exists a unique principal logarithms, denoted by 𝐿𝑜𝑔 𝐴, whose range lies in the infinite strip {𝑧 ∈ 𝐶: -𝜋 < 𝐼𝑚(𝑧) < 𝜋} of the Complex plane. Furthermore, for any given matrix norm‖. ‖,‖𝐴 -𝐼‖ < 1, where I be the identity transformation in 𝑅 𝑛 , then the series -∑

converges to 𝐿𝑜𝑔 𝐴 and consequently we can write

We denote that in general 𝐿𝑜𝑔(𝐴𝐵) ≠ 𝐿𝑜𝑔𝐴 + 𝐿𝑜𝑔𝐵. We here recollect the important fact by [Curtis, M.L. (1979)] [7] .

This result is also correct when 𝐿𝑜𝑔 in the above equation (3.5) is replaced by an analytic matrix function. So, we give the following essential theorem in the progress of our study. Proof: We recollect the following fats:

The facts (1), ( 2), ( 3), ( 4), ( 5) are easily verified.

For any real valued function f(x) (defined on a Riemannian manifold μ), the gradient ∇𝑓 is the unique tangent vector u at x such that

Where 〈𝑢, ∇𝑓〉, is the Riemannian inner product(R.I.P.) on the tangent space and 𝛾(𝑡) is a geodesic emanating from 𝑥 in direction of u.

A subset Ȃ of 𝜇(a Riemannian manifold) is convex if the shortest curve (geodesic) between any two points x and y is unique in μ and lies in Ȃ. A function (real valued) defined on a convex subset Ȃ of μ is convex of its limitations to any geodesic path is convex, i.e., 𝑖𝑓 𝑡 → 𝑓(𝑡) ≡ 𝑓(𝑒𝑥𝑝 𝑥 (𝑡𝑢)) is convex over its domain for all𝑥 ∈ 𝜇, is the exponential map at 𝑥. We prepared the space of all 𝑛 × 𝑛 symmetric matrices denoted by 𝑆(𝑛) = {𝐴 ∈ 𝜇(𝑛), 𝐴 𝑇 = 𝐴}, and The set of all 𝑛 × 𝑛 positive-definite matrices by ᴘ(𝑛) = {𝐴 ∈ 𝑆(𝑛), 𝐴 > 0}. Here 𝐴 > 0 means that matrix A is positive definite or the quadratic form 𝑥 𝑇 𝐴𝑥 > 0 for every 𝑥 ∈ 𝑅 𝑛 ̸ 0. It is well-known that ᴘ(𝑛) is an open convex cone or if 𝑃 𝑎𝑛𝑑 𝑄 are in ᴘ(𝑛), so is 𝑃 + 𝑡𝑄 for any 𝑡 > 0.

We remind that the exponential map from 𝑆(𝑛) to ᴘ(𝑛) is bijective (one-to-one and onto). In other words, positive-definite symmetric matrix is the exponential of symmetric matrix and the inverse of the exponential (i.e., the principal logarithms) for any positive-definite symmetric matrix is a symmetric matrix. ᴘ(𝑛) is an open subset of 𝑆(𝑛), for every 𝑃 ∈ ᴘ(𝑛), I classify the set 𝑇 𝑝 of tangent vectors to ᴘ(𝑛) at P with 𝑆(𝑛), on the Tangent Space at P, we define the positive-definite inner product and its corresponding norm as

which depends on the points P. The positive definiteness is a outcome of the positive definiteness of the Euclidean inner product for: We indicate that the length ℒ(𝛤) is invariant under congruent transformation, 𝛤 ⟼ 𝐶𝛤𝐶 𝑇 ,, for some fixed element C of 𝜖𝐺𝐿(𝑛). As 𝑑 𝑑𝑡 𝛤 -1 = -𝛤 -1 𝛤 ̇𝛤-1 , which implies that this length is also invariant under inversion.

In ᴘ(𝑛), the distance between two matrices A and B (considered as a differentiable manifold) is the infimum of lengths of curves joining them 1971)] [14] , is a (symmetric) matrix in the Tangent Space. Using invariance under congruent transformations, ᴘ(0) = 𝑃 and 𝑃 ̇(0) = 𝑆, the geodesic ᴘ(𝑛) is given by

It implies that the Riemannian metric of 𝑃 1 and 𝑃 2 in ᴘ(𝑛) is given by [16] .

We know that ᴘ(𝑛) is a homogeneous space of the Lie group 𝐺𝐿(𝑛) (where ᴘ(𝑛) given by the quotient

. It is also a non compact type symmetric space. [Terras, A. (1988)] [19] . We will also take the symmetric space of some special positive matrices:

This submanifold of ᴘ(𝑛)can also be written with the quotient

. Here 𝑆𝐿(𝑛) is the special linear group of all matrices from𝐺𝐿(𝑛), whose determinant is one. We conclude that 𝑆𝑃(𝑛) is a completely geodesic submanifold of 𝑃(𝑛) [Maab, H. (1971)] [14] . Now since 𝑃(𝑛) = 𝑆𝑃(𝑛) × 𝐼𝑅 + , 𝑃(𝑛) can be considered as a foliated manifold whose dimension minus (-) one leaves are isomorphic to 𝐻 𝑝 (the hyperbolic space), where𝑝 = , which is minimized. Therefore, the Euclidean mean will not be considered any more. The Riemannian mean of 𝑃 1 , 𝑃 2 , … , 𝑃 𝑚 may be called as the Riemannian barycenter of matrices𝑃 1 , 𝑃 2 , … , 𝑃 𝑚 , which concept was introduced by [Grove, Karcher and Ruh (1974)]. [Karcher, H. (1977)] [10,12] proved that for manifolds with Non-Positive sectional curvature, the Riemannian mean (barycenter) is unique. We will give a property of the Riemannian barycenter in the following theorem. ⁄ , and 𝑊 ∈ 𝑃(𝑛), is a constant matrix. Using equation (3.5) and few properties of the trace of a matrix, we have:

So, the result of Theorem 4.1, applied to the above gives

Hence, ∇𝐻 (the gradient of H) is given by

This is certainly in the tangent space, i.e. in 𝑆(𝑛).

Let G be the objective function for the minimization problem of the geometric mean of positive numbers given in equation (1.2). Furthermore, unlike the case of positive numbers where equation (4.7) gives to an explicit expression of the above geometric mean, in general, due to the asymmetric nature of 𝑝(𝑛), (4.3) can't be solved in closed from.

Conclusion

Above all properties are related to the geodesic reversing isometry in the symmetric space here taken by us. Hence, the concept of geometric mean, which is completely based on the Riemannian metric, can be used to describe the geometric mean on other symmetric spaces. Also various other properties of arithmetic mean, geometric mean, harmonic mean can be obtained for future work.

1. Alice M, Mond B, Pecaric J, Volenec V. Bounds for the matrix means SIAM J, Matrix Anal. Appl 1997;18:119-123.
2. Anderson WN, Trapp GE. Shorted operators, SIAM J. Appl. Math 1975;28:60-71.
3. Berger M, Gostiaux B. Differential Geometry: Manifolds, Curves, and Surfaces, Springer-Verleg, New York, 1998.
4. Bottema O. On the medians of a triangle in hyperbolic geometry, Canad. J Math, 1958, 502-506.
5. Bullen PS, Mitrinovic DS, Vasic PM, Means and their Inquelities, Mathematics and its Applications (East European Series), 31, D. Reidel Publishing Co. Dordrecht, The Netherlands, 1988.
6. Cowin SC, Yang G. Averaging anisotropic elastic constant data, J. Elasticity 1997;46:151-180.
7. Curtis ML. Matrix Groups, Springer-Verleg, New York-Heidelberg, 1979.
8. Eblerlein PB. Geometry of Nonpositively Curved Manifolds, The Jouns Hopkins University Press, London 1989.
9. Golub GH, Loan CFV. Matrix Computations, The Johns Hopkins University Press, London, 1989.
10. Grove K, Karcher H, Ruh EA. Jacobi fields and Finsler metrics on compact Lie groups with an applications to differentiable pinching problem, Math. Ann 1974;211:7-211.
11. Hardy GH, Lattlewood JE, Polya G. Inequalities, Cambridge University Press l London, 1989.
12. Karcher H. Riemannian center of mass and mollifier smoothing, Comm. Pure Appl. Math 1977;30:509-541.
13. Lawson JD, Lim Y. The geometric mean, matrices and more, Amer. Math. Monthly 2001;108:797-812.
14. Maab H. Siegel's Modular Forms and Dirichlet Series, Lecture Notes in Mathematics 216, springer-Verleg, Heidelerg, 1971.
15. Moakher M. Means and averaging in the group of rotations, SIAM J. Matrix Anal. Appl 2002;24:1-16.
16. Mostow GD. Strong Rigidity of Locally Symmetric Spaces, Annals of Mathematics Studies, Princeton University Press, Princeton, 1973.
17. Pusz W, Woronowicz SL. Punctional calculus for sesquilinear forms and the purification map, Rep. Math. Phy 1975;8:159170.
18. Spiesser M. Les mediates dans la pensee grecque, Musique et Mathematiques, Sci. Tech. Perspect 1993;23:1-71.
19. Terras A. Harmonic Analysis on Symmetric Spaces and Applications II, Springer-Verleg, New York, 1988.
20. Trapp GE. Hermitian semidefinite matrix means and related matrix inequalities-an introduction, Linear and Multilinear Algebra 1984;16:113-123.
21. Terras A. Convex Functions and Optimization Methods on Riemannian Manifolds. Mathematics and Its Applications Kluwer Academic, Dordcht, The Netherlands, 1988, 297.